Dependent Density Functional Theory...
137
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Transcript of Dependent Density Functional Theory...
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이학박사 학위논문
Mixed-Reference Spin-Flip Time-
Dependent Density Functional
Theory (MRSF-TDDFT): An accurate and efficient method of
calculating electronic excited states
for studying light-energy conversion
mechanisms
혼합기준 스핀젖힘 시간의존 밀도범함수 이론: 빛 에너지 전환 메커니즘 연구를 위한 정확하고
효율적인 전자 전이상태 계산 방법
2019년 8월
서울대학교 대학원
화 학 부
이 승 훈
Mixed-Reference Spin-Flip Time-
Dependent Density Functional Theory
(MRSF-TDDFT): An accurate and efficient
method of calculating electronic excited states for
studying light-energy conversion mechanisms
지도 교수 이 상 엽
이 논문을 이학박사 학위논문으로 제출함
2019년 8월
서울대학교 대학원
화학부 물리화학전공
이 승 훈
이승훈의 이학박사 학위논문을 인준함
2019년 8월
위 원 장 (인)
부위원장 (인)
위 원 (인)
위 원 (인)
위 원 (인)
감사의 글
어느 새 5년 반의 박사과정을 마치고 학위 논문을 제출하게
되었습니다. 많은 분들의 도움 덕분에 즐겁게 연구하며 졸업할 수
있었습니다. 학위 논문을 마치며 감사의 말씀을 전합니다.
한 없이 부족했던 제가 한걸음씩 성장해 나갈 수 있도록 인내심을
갖고 지도해 주시고, 항상 저를 믿고 아낌없이 지원 해주신
이상엽 교수님, 최철호 교수님께 진심으로 감사 인사 드립니다.
Many thanks to Dr. Michael and Dr. Hiroya for your kind help.
또한 따뜻한 격려와 조언을 해주신 석차옥 교수님, 정연준 교수님
그리고 성재영 교수님께도 감사 드립니다.
박사과정 동안 얻은 가장 큰 선물은 사랑하는 은지와 결혼한 것과
세상에서 가장 귀여운 아들 상백이가 태어난 것 입니다. 덕분에
너무 행복한 나날을 지낼 수 있었습니다. 감사합니다.
말로 표현을 많이 못해 드렸지만, 엄마 아빠 장모님 장인어른
항상 너무 감사하고 사랑합니다.
ABSTRACT The mixed-reference spin-flip time-dependent density functional
theory (MRSF-TDDFT) is proposed, which is derived from linear
response formalism for the time-dependent Kohn-Sham equation by
the use of mixed reference. Linear response from the mixed reference,
which combines MS = +1 and -1 components of triplet state, generates
additional configurations in the realm of TDDFT. Resultantly, MRSF-
TDDFT eliminates the erroneous spin-contamination of the SF-
TDDFT. Analytic energy gradients of the response states with respect
to nuclear coordinates are also derived and implemented. The
computational overhead of singlet or triplet states for MRSF-TDDFT
is nearly identical to that of SF-TDDFT. The resulting MRSF-TDDFT
computational scheme has several advantages before the conventional
SF-TDDFT. Linear-response equations for the singlet and triplet
responses are clearly separated. This considerably simplifies the
identification of the excited states, especially in the `black-box'' type
applications, such as the automatic geometry optimization, reaction
path following, or molecular dynamics simulations of the targeted
states. Accuracy of MRSF-TDDFT has been tested and verified in
various ways including vertical-excitation energy, singlet-triplet
energy gap, adiabatic-excitation energy, optimized structure,
minimum energy conical intersection, nonadiabatic coupling term, and
nonadiabatic molecular dynamic simulation. Therefore, it is highly
expected that the MRSF-TDDFT has advantages over SF-TDDFT in
terms of both practicality and accuracy.
Keyword: Photochemistry, Spin-flip time-dependent density
functional theory, Spin contamination problem, Conical intersection,
Nonadiabatic coupling matrix term, Nonadiabatic molecular dynamics.
Student Number: 2014-21231
TABLE OF CONTENTS
ABSTRACT
CONTENTS
LIST OF FIGURES
LIST OF TABLES
INTRODUCTION .................................................................... 1
THEORETICAL BACKGROUND ............................................. 6
1. Idempotency of reduced density matrix ...................... 6
2. Time-dependent Kohn-Sham equation ...................... 8
3. Linear-response theory .............................................. 9
3.1 Volterra expansion .............................................. 9
3.2 Connection between linear response of density
and one-electron excitation ............................... 11
3.3 Linear response equation ................................... 13
3.4 Electronic excitation energy ............................... 14
4. Spin-flip TDDFT ..................................................... 15
MIXED-REFERENCE SPIN-FLIP TDDFT ............................ 20
1. Mixed-reference reduced density matrix ................. 23
1.1 Definition of mixed-reference reduced density
matrix ................................................................ 23
1.2 Molecular orbital of mixed-reference reduced
density matrix .................................................... 23
1.3 Restoring idempotency of the mixed-reference
reduced density matrix ...................................... 25
2. Linear-response equation of mixed-reference spin-flip
TDDFT ..................................................................... 28
2.1 Definition of separated excitation amplitude ....... 31
2.2 Disentangling different MS response ................... 35
2.3 Recovery of one-to-one relation between
configuration and excitation amplitude ............... 40
2.4 Separating matrix equations for singlet and triplet
response states .................................................. 48
2.5 Expectation value of S2 operator ........................ 51
2.6 Dimensional transformation matrix ................... 54
3. Spin-pairing coupling .............................................. 57
ANALYTIC ENERGY GRADIENT OF MIXED-REFERENCE SPIN-
FLIP TDDFT ........................................................................ 60
1. Lagrangian ............................................................... 61
2. Orbital stationary condition I (Coupled perturbed
Hartree-Fock equation): Lagrange multiplizer Z .... 62
3. Orbital stationary condition II: Lagrange
multiplizer W ........................................................... 65
4. Analytic energy gradient with respect to nuclear
degree of freedom .................................................. 67
NUMERICAL RESULTS ....................................................... 70
1. Vertical excitation energy ..................................... 71
1.1 P excited state of Be atom ................................. 71
1.2 Singlet valence excited state of molecule ........... 75
2. Singlet-triplet energy gap ..................................... 78
3. Geometry-optimization structure ........................... 81
4. Adiabatic excitation energy ................................... 85
5. Conical intersection ............................................... 87
5.1 Minimum energy conical intersection of PSB3 .... 87
5.2 Topology of conical intersection for PSB3 .......... 89
6. Non-adiabatic coupling term .................................. 91
6.1 Numerical non-adiabatic coupling term .............. 91
6.2 Fast overlap evaluations with truncation ............. 92
6.3 Accuracy of truncation ....................................... 95
7. Non-adiabatic molecular dynamics ........................ 98
7.1 Photoisomerization and photocyclization of
cis-stilbene ....................................................... 98
7.2 Branching ratio and quantum yield .................... 100
7.3 Dynamics in the branching points ..................... 104
CONCLUSION .................................................................... 109
REFERENCES .................................................................... 112
ABSTRACT IN KOREAN ................................................. 124
LIST OF FIGURES
Figure 1. A schematic diagram of SF-TDDFT ....................... 19
Figure 2. A schematic diagram of MRSF-TDDFT ................. 22
Figure 3. Occupation of open-shell orbitals in a single KS
determinant and its relationship with the respective zeroth-
order RDM ............................................................................ 26
Figure 4. Type I configurations and notation of their Slater
determinants originating from spin flip transitions from the MS =
1 and -1 components of mixed reference ........................... 41
Figure 5. Connection between one electron transition from the
mixed-spin reference and that from the open-shell
configuration ......................................................................... 42
Figure 6. Comparison of the absolute errors (AE) of singlet-
triplet energy gaps for the organic molecules. ................... 80
Figure 7. Geometric RMSDs of SF-, MRSF(0)- and MRSF-TDDFT
with respect to the reference geometry of EOM-CCSD ........ 84
Figure 8. Magnitudes of AEE differences of SF-, MRSF(0)- and
MRSF-TDDFT with respect to the reference AEE of EOM-CCSD
in eV ...................................................................................... 86
Figure 9. Aligned MECI geometries for the PSB3 molecule
optimized by MRSF-, MRSF(0)-, SF-TDDFT, and MR-CISD 88
Figure 10. Energy difference between S1 and S0 states calculated
a loop around the conical intersection of PSB3 molecule with
MRSF-TDDFT. .................................................................... 90
Figure 11. Ratios of computation times for evaluating OIs varying
the truncation order with respect to the time with the lowest
truncation order. ................................................................... 94
Figure 12. Schematic diagram of cis-stilbene photodynamics
99
Figure 13. Two-dimensional histogram of cis-stilbene NAMD
dynamics simulation ............................................................. 107
Figure 14. Averaged <S2> value of SF-TDDFT for 50
trajectories in excited states ............................................... 108
LIST OF TABLES
Table 1. Connection between the excitation amplitudes of the SF-
TDDFT from each MS = +1 or -1 reference, and the new
excitation amplitudes of the MRSF-TDDFT from the mixed
reference ............................................................................... 30
Table 2. Detailed connection between the new excitation
amplitudes and their respective separated contributions ...... 33
Table 3. Details of the block-coupling matrices .................... 39
Table 4. Ground state total energies (Hartree) and excitation
energies (eV) for Be atom ..................................................... 74
Table 5. Mean absolute errors MAE (in eV) for the vertical
excitation energies computed at MRSF-, SF-, and LR-TDDFT
level of theory compared against the TBE-2 reference ........ 77
Table 6. Errors of NACT with the truncated OIs ................... 97
Table 7. Branching ratio (%) at three branching points during non-
adiabatic molecular dynamics of cis-stilbene ...................... 102
Table 8. Quantum yields (%) of ππ* excited cis-stilbene .... 103
１
INTRODUCTION
Proper and efficient descriptions of electronic excited states have
become more important than ever, as the theoretical studies of
emerging sciences, such as photovoltaics,[1, 2] molecular rotor[3]
are heavily dependent on them. A widely used methodology for
studying molecular excited states is the linear-response time-
dependent density functional theory (LR-TDDFT).[4-10] It is based
on the time-dependent Kohn-Sham (TD-KS) equation with the linear-
response formalism using a singlet ground state as a reference state.
In this approach, a fictitious noninteracting system is introduced whose
density is equal to that of the real interacting system.[11, 12] A wave
function of the noninteracting system is assumed to be described by a
single Slater determinant, which is referred to as an idempotency of
reduced density matrix in terms of the density matrix.[13, 14]
Contrasting to its popularity, well-known failures of this
methodology exist, in describing the energy of long-range charge
transfer excitations,[15-19] excited states with substantial double
excitation character,[20-23] excited states of molecules undergoing
bond breaking,[23-25] and real and avoided crossings between the
ground and excited states of molecules.[26-29] Some of these
２
drawbacks, in particular, the incorrect description of the S1/S0 conical
intersections and the poor description of multi-reference electronic
states, can be corrected to some extent by the spin-flip (SF)-
TDDFT,[30-32] which employs an Ms = +1 component of triplet
ground state e.g., αα , as a reference state instead of the closed-
shell reference of the LR-TDDFT. However, the use of only one
component of the degenerate triplet state leads to considerable spin
contamination of the resulting excited electronic states, except in a
few low-lying excited states.[33] The spin contamination of SF-
TDDFT is different from that of the unrestricted Hatree-Fock (UHF)
wavefunction come from orbital asymmetries. On the other hand, in
SF-TDDFT, the main source of spin contamination is due to spin
incompleteness of an excited set of configurations. Therefore, a key
solution for this problem is to expand the response space of SF-
TDDFT such that it can include the missing configurations.
Within the same contexts, several approaches have been
developed to tackle the spin-contamination problem of the SF
configuration interaction with single excitations (CIS),[33-37] which
is a wave function version of SF-TDDFT. However, unlike SF-CIS, a
considerable challenge remains with respect to TDDFT when going
beyond the adiabatic approximation to account for more than single
３
excitations.[38, 39] Because of this difficulty, only a few methods
have been developed to address the spin-contamination problem of
the SF-TDDFT. One means of adding more responses is to use a
higher excitation operator.[38] Without the TD-KS equation being
utilized, on the other hand, tensor equation-of-motion (TEOM)
approaches achieved considerable progress, yielding a series of SA-
SF-DFT methodologies.[36, 40, 41] These approaches can produce
correct spin eigenstates by applying tensor operators to a tensor
reference. However, the matrix elements of TEOM are evaluated using
the Wigner-Eckart theorem, which is not satisfied by the approximate
density functionals. Thus, the SA-SF-DFT formalism[36] requires an
a posteriori DFT correction to the SA-SF-CIS equations. Due to the
complexity of TEOM, the analytic energy gradient for the SA-SF-DFT
has yet to be derived.
Rather than using a high excitation operator[38] or using a
tensor operator with a tensor reference,[36] a means of expanding
the response space by linear response from more than one
reference[42] is shown in this thesis. In THEORETICAL
BACKGROUND section, a brief review of derivation for linear
response equation from time-dependent Kohn-Sham equation is
described. In addition, SF-TDDFT is introduced and its advantages
４
and disadvantages are summarized. In MIXED-REFERENCE SPIN-
FLIP TDDFT section, mixed-reference spin-flip (MRSF)-TDDFT is
proposed which eliminate spin-contamination problem of SF-TDDFT
while maintaining its advantages. Linear response equation of MRSF-
TDDFT is derived and a posteriori coupling, which is referred as spin-
pairing coupling, is introduced. In ANALYTIC ENERGY GRADIENT OF
MIXED-REFERENCE SPIN-FLIP TDDFT section, analytic energy
gradient is derived by the Lagrangian of MRSF-TDDFT. In
NUMERICAL RESULTS section, accuracy of MRSF-TDDFT has been
tested and verified in various ways including vertical-excitation
energy, adiabatic-excitation energy, optimized structure, minimum
energy conical intersection, nonadiabatic coupling term, and
nonadiabatic molecular dynamic simulation.
It is noted that this thesis reorganizes the contents of three papers
published in international journals and those of papers in preparation.
MIXED-REFERENCE SPIN-FLIP TDDFT
S. Lee, M. Filatov, S. Lee, C.H. Choi, J. Chem. Phys. 149, 104101 (2018)
ANALYTIC ENERGY GRADIENT OF MIXED-REFERENCE SPIN-FLIP TDDFT
S. Lee, E.E. Kim, H. Nakata, S. Lee, C.H. Choi, J. Chem. Phys. 150, 184111 (2019)
NUMERICAL RESULTS
1. Vertical excitation energy
S. Lee, M. Filatov, S. Lee, C.H. Choi, J. Chem. Phys. 149, 104101 (2018)
５
Y. Horbatenko and S. Lee, M. Filatov, C.H. Choi submitted for publication
2. Singlet-triplet energy gap
Y. Horbatenko and S. Lee, M. Filatov, C.H. Choi submitted for publication
3. Geometry-optimization structure
S. Lee, E.E. Kim, H. Nakata, S. Lee, C.H. Choi, J. Chem. Phys. 150, 184111 (2019)
3. Adiabatic excitation energy
S. Lee, E.E. Kim, H. Nakata, S. Lee, C.H. Choi, J. Chem. Phys. 150, 184111 (2019)
4. Conical intersection
S. Lee, E.E. Kim, H. Nakata, S. Lee, C.H. Choi, J. Chem. Phys. 150, 184111 (2019)
S. Lee and S. Shostak, M. Filatov, C.H. Choi, submitted for publication
5. Non-adiabatic coupling matrix elements
S. Lee, E.E. Kim, S. Lee, C.H. Choi, J. Chem. Theory Comput. 15, 882 (2019)
6. Non-adiabatic molecular dynamics
S. Lee, E.E. Kim, M. Filatov, S. Lee, C.H. Choi, in prepararation
６
THEORETICAL BACKGROUND
1. Idempotency of reduced density matrix
The essence of time-dependent density functional theory (TDDFT) is
the determination of the exact density in a time domain or in a
frequency domain from which any many-particle observable can be
obtained.[4-10] In this paper we shall only consider the density of
electrons. Based on the Runge-Gross theorem, the fictitious non-
interacting Kohn-Sham (KS) system is defined.[11, 12] In this system,
there is no interaction between electrons but all potentials are
described as one-particle operators. An one-electron wave function
of non-interacting KS system, ( , )ii t xσψ , is referred as KS molecular
orbital (MO). The i and iσ stand for index of the spatial part and
that of spin part for the i th occupied KS MO, respectively, and x
denotes both position and spin of the electron, ( , )x r σ= . A many-
electron wave function of the non-interacting system, [ ]( )tρΨ , is
usually restricted to be a normalized single Slater determinant. Then,
the reduced density matrix (RDM) of the [ ]( )tρΨ can be represented
as
occ
( , , ) ( , ) ( , )i i
i
i ii
t x x t x t xσ σ
σ
ρ ψ ψ′ ′=∑ , (1)
７
where the summation index of iiσ denotes a summation over occupied
MOs. It can be rewritten with the time-independent KS MO, ( )pp xσφ ,
(which is the solution of the usual time-independent KS equation) as
( , , ) ( ) ( ) ( )p q
p q
p q
p q p qp q
t x x P t x xσ σσ σ
σ σ
ρ φ φ′ ′= ∑ (2)
where summation indices of ppσ and qqσ represent summations
over whole MOs, and ( )p qp qP tσ σ is the discrete representation of the
RDM; denoted as the density matrix, in the following. The diagonal
part of the RDM is the density of the non-interacting system,
( , ) ( , , ),t x t x xρ ρ= (3)
which is assumed to be equal to the density of the corresponding real
system. In this paper, we shall only consider a case of systems
consisting of even-number (2n) electrons,
Tr ( , , ) 2 .t x x nρ = (4)
Due to the restriction of single Slater determinant of [ ]( )tρΨ , the
RDM is idempotent,
( , , ) ( , , ) ( , , ) ,t x x t x x t x x dxρ ρ ρ′ ′′ ′′ ′ ′′= ∫ (5)
which can be rewritten for the density matrix as
( ) ( ) ( ), , .p q p t t q
t
p q p t t q p qt
P t P t P t p qσ σ σ σ σ σσ
σ σ= ∀∑ (6)
８
2. Time-dependent Kohn-Sham equation
The TD density can be determined by the TD-KS equation which will
be discussed in this subsection. First, let us consider the Dirac action:
2
0[ ] [ ]( ) ( ) [ ]( ) .
tA t i H t t dt
tρ ρ ρ∂
= Ψ − Ψ∂∫ (7)
The stationary action principle with respect to the KS MO,
[ ] ( , ) 0iiA t xσδ ρ δ ψ = , leads to the TD-KS equations,[43]
( , ) [ ( )] ( , ) , occupied MO.i ii i ii t x F t t x i
tσ σψ ρ ψ σ∂
= ∈∂
(8)
In terms of the RDM, the TD-KS equation can be rewritten as
,i F Ftρ ρ ρ∂= −
∂ (9)
which become
( ) , , whole MO.p q p t t q p t t q
t
p q p t t q p t t q p qt
i P F P P F p qt σ σ σ σ σ σ σ σ σ σ
σ
σ σ∂= − ∈
∂ ∑ (10)
where
* 2 ( , )1 [ ]( ) ( ) .2 ( , )
p q
p q
XCA
p q p qA A
Z t x AF x dx x dxr R r r t x
σ σσ σ
ρ δ ρφ φδρ
′′= − ∇ − + + ′− −
∑∫ ∫ (11)
Within the adiabatic approximation, the exchange-correlation (XC)
part XCA of the action functional is replaced by the XC functional of
time-independent DFT evaluated with the density tρ at time t,
[ ] ( , ) [ ] ( )XC XCt tA t x E xδ ρ δρ δ ρ δρ≅ . In the case of approximate hybrid XC
９
functionals, the Fock matrix, p qp qF σ σ , becomes
* 2
*
* *
1( ) ( )2
[ ](1 ) ( ) ( )
( )ˆ(1 ( , ))
( ) ( ) ( ) ( ) ( ) ,
p q
p q
p q
p qs r
r s
r s
Ap q p q
A A
XCt
H p qt
Hr s p s r q
r s
ZF x x dx
r R
Ec x x dx
x
c P x xP t x x x x dxdx
r r
σ σσ σ
σ σ
σ σσ σσ σ
σ σ
φ φ
δ ρφ φ
δρ
φ φ φ φ
= − ∇ − −
+ −
′−′ ′ ′+
′ −
∑∫
∫
∑ ∫ ∫
(12)
where ˆ( , )P x x′ is the permutation (exchange) operator and Hc is the
mixing coefficient for the exact (Hartree-Fock, HF) exchange; Hc =
0 or 1 recovers the pure DFT or pure HF limits, respectively.
3. Linear-response theory
A way to solve the density-matrix formulation of TD-KS equation in
Eq. (10) within linear-response formalism is suggested by using the
idempotency of RDM in Eq. (6).[13] In this subsection, a review of the
derivation of linear-response equation is presented starting from the
Volterra expansion.
3.1 Volterra expansion
Suppose a time-dependent one-electron external perturbation, e.g., a
time-dependent electric field, with a frequency Ω and a strength λ
１０
ext ( ) c.c.p q p q
i tp q p qv t h eσ σ σ σλ − Ω= + (13)
is applied to a reference system at time 0, whose density can be
determined by the usual time-independent DFT. Here, c.c. denotes
complex conjugate of the preceding term. Then, the density matrix,
( )p qp qP tσ σ , and the Fock matrix, ( )
p qp qF tσ σ , can be expanded in powers
of λ as
(0) (1) 2( ) ( ) ( ),p q p q p qp q p q p qP t P P t Oσ σ σ σ σ σλ λ= + + (14)
(0) (1) 2( ) ( ) ( ).p q p q p qp q p q p qF t F F t Oσ σ σ σ σ σλ λ= + + (15)
If the reference system is described by a single Slater determinant,
the zeroth-order density and Fock matrices are given in
(0), for , occupied MO,
otherwise, 0p q
p q
pq p qp q
p qP σ σ
σ σ
δ δ σ σ ∈=
(16)
*(0) 2
* 0
0
occ* *
1( ) ( )2
[ ](1 ) ( ) ( )
( )ˆ(1 ( , ))
( ) ( ) ( ) ( ) ,
p q
p q
p q
p qi i
i
Ap q p q
A A
XC
H p q
Hp i i q
i
ZF x x dx
r R
Ec x x dx
x
c P x xx x x x dxdx
r r
σ σσ σ
σ σ
σ σσ σ
σ
φ φ
δ ρφ φ
δρ
φ φ φ φ
= − ∇ − −
+ −
′−′ ′ ′+
′ −
∑∫
∫
∑∫ ∫
(17)
The first-order density and Fock matrices can be represented by
(1) ( ) c.c.,p q p q
i tp q p qP t d eσ σ σ σ
− Ω= + (18)
(1) ext (1)( ) ( ) ( ).p q
p q p q r s
r s r s
p qp q p q r s
r s r s
FF t v t P t
Pσ σ
σ σ σ σ σ σσ σ σ σ
∂= +
∂∑ (19)
The matrix, p qp qd σ σ in Eq. (18), represent amplitudes of linear
１１
response of density. The first and the second terms of Eq. (19) are the
external perturbation potential and the response of the Fock matrix to
the density variation, respectively.
3.2 Connection between linear response of density and one-electron excitation
Substituting expansion of density matrix of Eq. (14) into the
idempotency relation of Eq. (6) yields the idempotency relations for
the zeroth- and first-order density matrices, respectively,
(0) (0) (0) ,p t t q p q
t
p t t q p qt
P P Pσ σ σ σ σ σσ
=∑ (20)
( )(1) (0) (0) (1) (1) , , .p t t q p t t q p q
t
p t t q p t t q p q p qt
P P P P P p qσ σ σ σ σ σ σ σ σ σσ
σ σ+ = ∀∑ (21)
The former relation of Eq. (20) satisfied with the condition in Eq. (16)
enable to define projection matrices onto the subspace of the occupied
MOs, (0)i ii iPσ σ , and onto the unoccupied MOs, (0)
a a a aa a a aPσ σ σ σδ − . The orbital
index convention used in this section is i, j for occupied MOs, a, b for
virtual MOs, p, q, r, s, t for MOs in general.
Meanwhile, substituting the first-order density matrix of Eq. (18) into
the first-order idempotency relation of Eq. (21) yields
( )(0) (0) ,p t t q p t t q p q
t
p t t q p t t q p qt
d P P d dσ σ σ σ σ σ σ σ σ σσ
+ =∑ (22)
１２
from which the amplitude matrix p qp qd σ σ can be represented as
,p q p q p qp q p q p qd X Yσ σ σ σ σ σ= + (23)
where
(0) (0)( ) ,
p q p r p r r s s q
r s
p q p r p r r s s qr s
X P M Pσ σ σ σ σ σ σ σ σ σσ σ
δ= −∑ (24)
(0) (0)( ).p q p r r s s q s q
r s
p q p r r s s q s qr s
Y P M Pσ σ σ σ σ σ σ σ σ σσ σ
δ= −∑ (25)
Here r sr sM σ σ is an arbitrary matrix.
p qp qX σ σ has non-zero elements
between the unoccupied ppσ and occupied qqσ MOs and
corresponds to one-electron excitations, whereas the matrix p qp qY σ σ
has non-zero elements for the occupied ppσ and unoccupied qqσ
MOs and corresponds to one-electron de-excitations. It is emphasized
that the first-order density matrix can be represented in terms of MOs
of reference system, and these can be interpreted as one-electron
excitation and de-excitation from the Eqs. (24) and (25).
In this paper we shall use Tamm-Dancoff approximation (TDA)[44, 45]
in which the excitation amplitude, p qp qX σ σ , is only considered and the
de-excitation amplitude p qp qY σ σ is ignored.
１３
3.3 Linear response equation
Substituting expansion of density and Fock matrices of Eqs. (14) and
(15) into the TD-KS equation of Eq. (10) yields equation of motions
for the nth-order density matrix. These for the zeroth- and the first-
order density matrix are given by
( )(0) (0) (0) (0)0 ,p t t q p t t q
t
p t t q p t t qt
F P P Fσ σ σ σ σ σ σ σσ
= −∑ (26)
( )(1) (0) (1) (1) (0) (1) (0) (0) (1) .p q p t t q p t t q p t t q p t t q
t
p q p t t q p t t q p t t q p t t qt
i P F P P F F P P Ft σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ
σ
∂= − + −
∂ ∑ (27)
The former relation of Eq. (26) is already satisfied with a condition in
Eq. (16), while the latter relation of Eq. (27) will determine linear-
response of density. Substituting the first-order density and Fock
matrice of Eqs. (18) and (19) into the the latter relation yields
( )(
(0) (0)
(0) (0)
(0) (0) .
p q p t t q p t t q
t
p t t q p t t q
t
p t t q
r s t q p t r s
r s r sr s r s
p q p t t q p t t qt
p t t q p t t qt
p t t qr s t q p t r s
r s r sr s r s
d h P P h
F d d F
F Fd P P d
P P
σ σ σ σ σ σ σ σ σ σσ
σ σ σ σ σ σ σ σσ
σ σ σ σσ σ σ σ σ σ σ σ
σ σ σ σσ σ σ σ
Ω = −
+ −
∂ ∂ + − ∂ ∂
∑
∑
∑ ∑
(28)
Within TDA, the right-multiplication of all terms in Eq. (28) by
projection matrices on occupied MO space, (0)q iq iPσ σ , and the left-
multiplication by projection matrices on virtual MO space,
(0)q p q pa p a pPσ σ σ σδ − , gives
１４
( ), ,a i a i b j i j a b b j
b j
a i a i b j ij ab b jb j
h A Xσ σ σ σ σ σ σ σ σ σ σ σσ σ
δ δ δ δ− = − Ω∑ (29)
where orbital Hessian matrix is
(0) (0), .a i
a i b j a b i j i j a b
j b b j
a ia i b j a b ij i j ab
j b b j
FA F F
Pσ σ
σ σ σ σ σ σ σ σ σ σ σ σσ σ σ σ
δ δ δ δ∂
= − +∂∑ (30)
3.4 Electronic excitation energy
Electronic excitation energies can be obtained from the linear-
response amplitude of density by analysis of the poles of the
polarizability suggested by M. E. Casida.[5] Within the sum-over-
states (SOS) approach, the dynamic polarizability is represented as
ref ref ref ref
,I I I I
x y x yxy
I II I
µ µ µ µα = +
Ω − Ω Ω + Ω∑ ∑ (31)
where ref ref
Ir Irµ = Ψ Ψ and IΩ are transition dipole moments in the
r direction and the excitation energy, respectively, from the reference
state to the Ith excited state, and Ω is frequency of the external
perturbation.
Meanwhile, a linear-response amplitude of the dipole moment in x
direction, xδµ , can be represented as
2 .q p p q
p q
x q p p qp q
x dσ σ σ σσ σ
δµ = − ∑ (32)
Thus, an element of polarizability is given by
１５
2 .p q
q p
p q
p qxxy q p
p qy y
dx σ σ
σ σσ σ
δµα = = − ∑E E
(33)
where yE is the TD perturbed electric field in y direction. As
mentioned in a previous section, amplitude of linear-response density,
p qp qd σ σ , is equal to the excitation amplitude, p qp qX σ σ , within TDA.
With Eq. (29), it can be rewritten as
( ) 1
,2 .q p p q r s i j a b r s
p q r s
xy q p p q r s ij ab r sp q r s
x A yσ σ σ σ σ σ σ σ σ σ σ σσ σ σ σ
α δ δ δ δ−
= − Ω∑ ∑ (34)
The first term on the right-hand side of Eq. (31) has poles at excitation
energies, IΩ , and the second term has poles at de-excitation energies,
I−Ω . Therefore, the excitation energy can be obtained by comparing
the first term of Eq. (31) and Eq. (34) within TDA.[44, 45] It follows
that the excitation energies are the solutions of the eigenvalue
problem,
, .p q r s r s p q
I Ip q r s r s I p qA X Xσ σ σ σ σ σ σ σ= Ω (35)
The final resulting equation is called as the linear-response equation
or the Casida equation.
１６
4. Spin-flip TDDFT
If the zeroth-order (reference) density matrix for the Volterra
expansion in Eq. (16) satisfy that of the closed-shell ground singlet
state, the linear-response equation of Eq. (35) is the conventional
linear-response TDDFT with TDA. Meanwhile, for the spin-flip (SF)-
TDDFT utilize the density matrix of the triplet open-shell
reference[30, 31] shown in upper panel of Fig. 1.
For consistency's sake of describing SF-TDDFT and mixed-reference
(MR) SF-TDDFT, notations used in this rest of thesis are redefined.
Indices for doubly and singly occupied KS spin molecular orbitals (MOs)
of reference states are labeled as i, j and x, y, respectively, whereas
those of the virtual KS spin MOs are labeled as a, b. Those for arbitrary
(occupied or virtual) KS spin MOs are written as p, q, r, s, t, u, and
four Greek indices (μ, ν, κ, λ) denote atomic orbitals. The σ and τ denote
the index for the spin function of the MO. In addition, the closed, open,
and virtual orbital spaces are denoted as C, O and V, respectively. The
number of electrons is 2n, thus the nth and the (n+1)th MOs
representing the two orbitals in O space. Indices of O1 and O2 are used
instead of n and n+1 for these two specific orbitals. To prevent
１７
repeated equations, the O1 and O2 are usually represented as Om (m=1,
2) or On (n=1, 2), respectively. The singlet and triplet states are
denoted as S and T, respectively, and the index for these is labeled as
k (k=S, T). All two-electron integrals are written in chemist notation.
Then, the density matrix of the MS = +1 triplet reference can be
represented as
(0) (0) (0)O1 O1 O2 O2= 1, = 1, = 1, otherwise 0.i ii i
P P Pσ σ α α α α (36)
The SF-TDDFT usually utilize TDA and collinear approximation for
exchange-correlation kernel. Thus, the linear-response equation of
SF-TDDFT is given by
, = ,I Ip q r s r s I pr qs r s
rs rsA X Xβ α β α β α β αδ δΩ∑ ∑ (37)
with
( )(0) (0), = | .p q r s p r sq s q pr HA F F c pr sqβ α β α β β α αδ δ− − (38)
Schematic diagram describing electronic configurations by one-
electron spin-flip excitation (linear response) from the triplet
reference is shown with black full arrows in Fig. 1. The configurations
are categorized by different initial and final MOs of a spin-flip
excitation for four types: TYPE I (O→O), TYPE II (C→O), TYPE III
(O→V), TYPE IV (C→V). The configurations shown by the gray dashed
arrows are missing in SF-TDDFT. Hence, the excitation space defined
１８
by SF-TDDFT is incomplete and spin-contaminated, as the individual
configurations shown in Fig. 1 are not eigenfunctions of the total spin
S2.
SF-TDDFT is known to have great advantages in describing single-
bond breaking or single-bond twisting systems.[30, 33] Two features
of SF-TDDFT can be supported this statement. One is that triplet
reference can well-describe two degenerate open-shell MOs, which
frequently occur when a bond is breaking or twisted. In other word,
SF-TDDFT can take into account static correlation. In addition, since
all singlet states are described as response states, there exists
coupling between ground and excited states.[46] It has been found
that there is a great advantage in describing avoided crossing or
conical intersection topology correctly.[27, 46] Despite of such many
advantages, severe spin-contamination of response states except for
few of low-lying excited states makes it difficult to use SF-TDDFT
practically.
１９
Figure. 1. A schematic diagram of SF-TDDFT. The upper panel depicts the
high-spin triplet reference and the corresponding zeroth-order RDM. In the
lower pannel, a complete set of electronic configurations considered in SF-
TDDFT is given. Electronic configurations which can be generated by SF
linear responses (SF one-electron transitions) from the zeroth-order RDM
are given by black arrows in four types. Configurations unable to be obtained
in the linear responses of SF-TD-DFT are given by gray dashed arrows.
２０
MIXED-REFERENCE SPIN-FLIP TDDFT
The main sources of spin-contamination in SF-TDDFT is the missing
electronic configurations (and the respective amplitudes) shown by the
gray arrows as type II, III and IV in Fig. 1. In this thesis, the new mixed
zeroth-order RDM is introduced[42] as an equiensemble of the MS =
+1 and -1 components of the triplet state in the first subsection of
Mixed-reference reduced density matrix. As shown in Fig. 2, the use
of the mixed-reference reduced density matrix (MR-RDM) includes
many of the electronic configurations missing in SF-TDDFT. Linear-
response equation with the MR-RDM is derived in the second
subsection of in the second subsection of Linear-response equation of
mixed-reference spin-flip TDDFT. A posteriori coupling between
configurations originating from MS = +1 and -1 are introduced in the
next subsection of Spin-pairing coupling. Furthermore, expectation
value of S2 operator is evaluated for the response states of MRSF-
TDDFT in the last subsection of Expectation value of S2 operator.
From this, it is proved that the spin-contamination of SF-TDDFT is
nearly eliminated in MRSF-TDDFT.[42]
It is noteworthy that not all of the electronic configurations shown in
Fig. 2 can be recovered by the use of the MR-RDM. Thus, four out of
２１
six type IV configurations (four configurations shown with the gray
arrows in Fig. 2) are still unencountered. Typically, these
configurations represent high lying excited states and make
insignificant contributions to the low lying states of molecules. Hence,
the effect of the missing configurations for the spin-contamination of
the SF-TDDFT response states is expected to be relatively small.
２２
Figure. 2. A schematic diagram of MRSF-TDDFT. The upper panel shows the zeroth-order MR-RDM which is a combination of MS = +1 and -1 RDMs. In the lower pannel, electronic configurations which can be generated by spin-flip linear responses from the MR-RDM are given by black arrows in four types. Configurations unable to be obtained in the linear responses of MR-SF-TD-DFT are given by gray dashed arrows.
２３
1. Mixed-reference reduced density matrix
1.1 Definition of mixed-reference reduced density matrix
The proposed mixed-reference reduced density matrix (MR-RDM) is
MR Ms=+1 Ms=-10 0 0
1( , ) ( , ) ( , ) .2
x x x x x xρ ρ ρ′ ′ ′= + (39)
In terms of the zeroth-order Kohn-Sham (KS) molecular orbital (MO),
it is represented by
*M * *
0 O1 O1 O1 O11 1( , ) = ( ) ( ) ( ) ( ) ( ) ( )2 2
R i ii i
i i
x x x x x x x xσ σ α α β β
σ
ρ φ φ φ φ φ φ′ ′ ′ ′+ +∑
* *O2 O2 O2 O2
1 1( ) ( ) ( ) ( ),2 2
x x x xα α β βφ φ φ φ′ ′+ + (40)
where the first term on the right hand side of the equation represents
the RDM of the C space, the remaining terms - the RDM of the O space.
The zeroth-order MR-RDM is a diagonal matrix with the elements
(0)M (0)M (0)M (0)M (0)M
O1 O1 O1 O1 O2 O2 O2 O21 1 1 1= 1, = , = , = , = ,otherwise0,2 2 2 2
R R R R Ri ii i
P P P P Pσ σ α α β β α α β β (41)
and its trace satisfy the number of electrons:
(0)MR 2 .p p
p
p pp
P nσ σσ
=∑ (42)
1.2 Molecular orbital of mixed-reference reduced density matrix
According to the ensemble DFT,[47, 48] the energy of MR-RDM of
２４
Eq. (39) is given by
( )= 1 = 1M0 0 0
1[ ] = [ ] [ ] ,2
M MR S SE E Eρ ρ ρ+ −+ (43)
where = 1
0[ ]MSE ρ + and
= 10[ ]MSE ρ −
are energies of = 1SM + and 1−
references, respectively. Since the energies of = 1SM + and 1−
references are same, the energy of MR-RDM and those of references
are same as
= 1 = 1M
0 0 0[ ] = [ ] = [ ].M MR S SE E Eρ ρ ρ+ − (44)
By the ensemble DFT, the spatial part of MOs consisting MR-RDM,
kφ , can be obtained with two conditions as
( )M0[ ] = 0,R
pq qp q ppqk
E Fδ ρ δ φ φδφ
+ −
∑ (45a)
*= ,pq qpF F (45b)
where pqF is the Lagrange multiplier. Substituting Eq. (44) into Eq.
(45a) yield
( )= 10[ ] = 0,MS
pq qp q ppqk
E Fδ ρ δ φ φδφ
+ + −
∑ (46a)
*= .pq qpF F (46b)
The conditions of Eqs. (46a) and (46b) are those of restricted open-
shell Kohn-Sham method for = 1SM + reference which is the way to
obtain the spatial part of MOs in SF-TDDFT. Therefore, one can use
same spatial part of MOs of SF-TDDFT in MRSF-TDDFT.
２５
1.3 Restoring idempotency of the mixed-reference reduced density matrix
The proposed MR-RDM does not satisfy the idempotency conditions
as in Eq. (20); e.g., an open-shell RDM of the new MR-RDM, (0)MR1 1O OP α α
(as well as (0)MR1 1O OP β β , (0)MR
2 2O OP α α , and (0)MR2 2O OP β β ), violates the condition
(0)MR (0)MR (0)MR (0)MR (0)MR (0)MRO1 O1 O1 O1 O1 O1 O1 O1 O1 O1
1 ,2t t
t
t tt
P P P P P Pα σ σ α α α α α α α α ασ
= = ≠∑ (47)
thus precluding straightforward derivation of the linear-response
equation. As follows from Eqs. (40) and (41), the non-idempotency of
the RDM of Eq. (47) originates from half-integer populations of the
zeroth-order open-shell KS orbitals O1σφ and O2
σφ , ,σ α β= . To
resolve this difficulty and to restore idempotency of the respective
RDM, we replace the original spin-orbitals (with the α or β spin) by
the orbitals of mixed spin, labeled in the following by s1 and s2,
obtained by the application of a unitary transformation U
1 2†
1 2
1 11 ,2 1 1
s s i i
s s i i
α α
β β
+ − = =
− + U (48)
which leads to the new mixed spin functions of the O space
1 2(1 ) (1 ) (1 ) (1 ), .
2 2i i i is sα β α β+ + − − + +
= = (49)
The new mixed spin O orbitals O , , 1, 2msk k mφ = are orthonormal among
themselves and with respect to the C and V orbitals.
２６
Figure. 3. Occupation of open-shell orbitals in a single KS determinant and its
relationship with the respective zeroth-order RDM. A single KS determinant shown in subfigure (a) represents the RDM of the SF-TDDFT method. Subfigure (b) shows equivalence between four possible occupation patterns of the mixed spin-orbitals s1 and s2, which all yield the same zeroth order RDM. Populations of the conventional α spin-orbitals are shown with upward pointing arrows and of the mixed spin-orbitals are shown with dots.
２７
There exist four possible choices for populating the new mixed spin-
orbitals, as shown in Fig. 3b. As all the four population patterns result
in the same RDM, for convenience, the configuration in Fig. 3b with
both s1 mixed spin-orbitals occupied is selected. With this choice of
occupations, the RDM becomes
1 1 1 1* * *MR0 O1 O1 O2 O2( , ) ( ) ( ) ( ) ( ) ( ) ( )i i
i
s s s si i
ix x x x x x x xσ σ
σ
ρ φ φ φ φ φ φ′ ′ ′ ′= + +∑ (50)
1 1 1 1
(0)MR (0)MR (0)MRO1 O1 O2 O2, 1, 1, otherwise 0.
i j i ji j ij s s s sP P Pσ σ σ σδ δ= = = (51)
With spatial part of MOs described by real function, the redefined RDM
of Eq. (50) is identical to Eq. (40), as can be easily seen by expanding
the open-shell contributions as, e.g.,
( ) ( )( ) ( ) ( )( )
1 1 2 1 1 2*O1 O1
* * * *O1 O1 O1 O1 O1 O1 O1 O1
* *O1 O1 O1 O1
( ) ( )1 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )21 ( ) ( ) ( ) ( )2
i i i ix x
x x x x i x x i x x
x x x x
α β α β
α α β β α β β α
α α β β
φ φ
φ φ φ φ φ φ φ φ
φ φ φ φ
+ + − + + − ′
′ ′ ′ ′= + + −
′ ′= +
(52)
and a similar expression for the other open-shell contribution to Eq.
(50). Replacing the old spin parts of the O spin-orbitals (represented
by the α and β spin functions) by the mixed spin functions s1 and s2,
while keeping the α and β spin functions for the C and V orbitals and
using Eq. (51), lets one to show that the overall idempotency relation
(0)MR (0) MR (0) MR .p t t q p q
t
p t t q p qt
P P Pσ σ σ σ σ σσ
=∑ (53)
holds for the redefined density matrix.
２８
2. Linear-response equation of mixed-reference spin-flip
TDDFT
As described by previous section, the idempotency of the zeroth-
order density matrix can be recovered by the introduced spin rotation.
Following same steps described in subsection 3 of the first
THEORETICAL BACKGROUND section one can straightforwardly
obtain a linear-response equation of MR-RDM. However, linear
responses from each of MS = +1 and -1 references should appear
mixed in the linear responses from the MR-RDM. Therefore, one need
to disentangle the mixed responses which is described in subsection
2.1 of Definition of separated excitation amplitude and subsection 2.2
of Disentangling different MS response. In addition, type I (O→O)
configurations shown in Fig. 2 can be generated by both MS = +1 and
-1 reference. Due to the unmatching of configuration from different
reference, additional rearrangement is required and it is presented in
subsection 2.3 of Recovery of one-to-one relation between
configuration and excitation amplitude. With these procedure, one can
obtain clearly separated linear-response equation for singlet and
triplet states, which is described in the next subsection 2.4 of
Separating matrix equations for singlet and triplet response states.
With dimensional transformation matrix introduced in the last
２９
subsection 2.5 of Dimensional transformation matrix, the linear-
response equation for the singlet and triplet states can be described
succinctly with a single form. Furthermore, this concept is useful for
the implementation of MRSF-TDDFT with a little modification of code
of SF-TDDFT.
３０
SF-TDDFT MRSF-TDDFT
= 1SM + = 1SM − Mixed Separated
reference reference amplitude amplitude IO CXβ α
+ IO CXα α
→ M ,
2
R IO Cs
Xα
→ =0,
2
M ISO Cs
Xα
+ = 1,
2
M ISO Cs
Xα
−
IO CXβ β
+ IO CXα β
→ M ,
2
R IO Cs
Xβ
→ =0,
2
M ISO Cs
Xβ
+ = 1,
2
M ISO Cs
Xβ
+
IV OXβ α
+ I
V OXβ β
→ M ,
1
R IV Os
Xβ
→ =0,
1
M ISV Os
Xβ
+ = 1,
1
M ISV Os
Xβ
−
IV OXα α
+ I
V OXα β
→ M ,
1
R IV Os
Xα
→ =0,
1
M ISV Os
Xα
+ = 1,
1
M ISV Os
Xα
+
IV CXβ α
+ I
V CXβ α
→ M ,R IV CXβ α
→ =0,M IS
V CXβ α
+ = 2,M IS
V CXβ α
−
IV CXα β
+ I
V CXα β
→ M ,R IV CXα β
→ =0,M IS
V CXα β
+ = 2,M IS
V CXα β
+
IO OXβ α
+ IO OXα β
→ M ,
2 1
R IO Os s
X → =0,
2 1
M ISO Os s
X + R ,
2 1
edun IO Os s
X
IV CXα α
+ I
V CXα α
→ M ,R IV CXα α
IV CXβ β
+ I
V CXβ β
→ M ,R IV CXβ β
Table 1. Connection between the excitation amplitudes of the SF-TDDFT from
each MS = +1 or -1 reference, and the new excitation amplitudes of the MRSF-
TDDFT from the mixed reference.
３１
2.1 Definition of separated excitation amplitude
As shown in Tab. 1, it is natural to think that total eighteen types of
independent excitation amplitudes p q
Ip qX σ σ from MS = +1 or MS = -1
reference, are entangled in the nine types of new excitation amplitudes
MR,p q
Ip qX σ σ ; for each row, two
p q
Ip qX σ σ are entangled in MR,
p q
Ip qX σ σ . It is noted
that 2 1
MR,O Os s
IX on the 7th row has a redundant part since the
configurations are same generated by O→O excitations from two
different references.
The starting point for separating these entangled excitation amplitudes
can be found in the general expression of the summation
( ) MR,p q r s r sr s
Ip q r s r sr s
F P Xσ σ σ σ σ σσ σ∂ ∂∑ in the matrix equation. For any index
ppσ and qqσ , the summation index rrσ and ssσ can be expanded
into the 7 types of Tab. 1 as
M , =p qp q R I
r sr sr s r sr s r s
FX
Pσ σ
σ σσ σ σ σ
∂
∂∑
M , M ,
2 2
1 1 1 12 2 2 2
p q p qp q p qR I R Iws j ws j
wj wjw j w j
F Fi iX XP Pσ σ σ σ
α αβ α α α
∂ ∂ + − + + ∂ ∂
∑ ∑ (54a)
M , M ,
2 2
1 1 1 12 2 2 2
p q p qp q p qR I R Iws j ws j
wj wjw j w j
F Fi iX XP Pσ σ σ σ
β βα β β β
∂ ∂ − + + + ∂ ∂
∑ ∑ (54b)
M , M ,
1 1
1 1 1 12 2 2 2
p q p qp q p qR I R Ib zs b zs
bz bzb z b z
F Fi iX XP Pσ σ σ σ
β ββ α β β
∂ ∂ − + + + ∂ ∂
∑ ∑ (54c)
３２
M , M ,
1 1
1 1 1 12 2 2 2
p q p qp q p qR I R Ib zs b zs
bz bzb z b z
F Fi iX XP Pσ σ σ σ
α αα β α α
∂ ∂ + − + + ∂ ∂
∑ ∑ (54d)
M , M ,1 1 1 1
2 2 2 2
p q p qp q p qR I R Ib j b j
bj bjb j b j
F FX X
P Pσ σ σ σ
β α β αβ α β α
∂ ∂ + + ∂ ∂
∑ ∑ (54e)
M , M ,1 1 1 12 2 2 2
p q p qp q p qR I R Ib j b j
bj bjb j b j
F FX X
P Pσ σ σ σ
α β α βα β α β
∂ ∂ + + ∂ ∂
∑ ∑ (54f)
M , M ,
2 1 2 1
1 1 .2 2
p q p q p q p qp q p q p q p qR I R Iws zs ws zs
wz wzw z w z w z w z
F F F FX iX
P P P Pσ σ σ σ σ σ σ σ
β α α β α α β β
∂ ∂ ∂ ∂ + + + − ∂ ∂ ∂ ∂ ∑ ∑ (54g)
In each of Eqs. (54a)-(54f), two terms are given come from MS = +1
and -1 references. The excitation amplitude in the left term
corresponds to a configuration with MS = 0, and that in the right term
corresponds to a configuration with MS ≠ 0. While, in Eq. (54g), the left
term come from both MS = +1 and -1 references, and the right term
is the redundant term. From this fact, the separated excitation
amplitudes can be defined as in Tab. 2.
３３
= 0SM amplitudes 0SM ≠ amplitudes
=0,
2
M ISus iX α ≡ M ,
2
12
R Ius i
i X α+
= 1,
2
M ISus iX α
− ≡ M ,
2
12
R Ius i
i X α−
=0,
2
M ISus iX α =
= 1,
2
M ISus iiX α
−
=0,
2
M ISus iX β ≡ M ,
2
12
R Ius i
i X β−
= 1,
2
M ISus iX β
+ ≡ M ,
2
12
R Ius i
i X β+
=0,
2
M ISus iX β = −
= 1,
2
M ISus iiX β
+
=0, M ,
1 1
12
M I R ISa vs a vs
iX Xβ β−
≡ = 1, M ,
1 1
12
M I R ISa vs a vs
iX Xβ β− +
≡ =0,
1
M ISa vsX β = −
= 1,
1
M ISa vsiX β
−
=0, M ,
1 1
12
M I R ISa vs a vs
iX Xα α+
≡ = 1, M ,
1 1
12
M I R ISa vs a vs
iX Xα α+ −
≡ =0,
1
M ISa vsX α =
= 1,
1
M ISa vsiX α
+
=0, M ,12
M I R ISa i a iX Xβ α β α≡
= 2, M ,12
M I R ISa i a iX Xβ α β α
− ≡ =0,M ISa iX β α =
= 2,M ISa iX β α
−
=0, M ,12
M I R ISa i a iX Xα β α β≡
= 2, M ,12
M I R ISa i a iX Xα β α β
+ ≡ =0,M ISa iX α β =
= 2,M ISa iX α β
+
=0, M ,
2 1 2 1
M I R ISus vs us vsX X≡
R , M ,
2 1 2 1
edun I R Ius vs us vsX iX≡ =0,
2 1
M ISus vsX = − R ,
2 1
edun Ius vsiX
Table 2. Detailed connection between the new excitation amplitudes and their
respective separated contributions. The MS value represented at superscript
of a newly defined excitation amplitude denotes the value of a corresponding
configuration.
３４
After substituting amplitude matrices defined in Tab. 2, the equation
54a-g becomes
M , =p qp q R I
r sr sr s r sr s r s
FX
Pσ σ
σ σσ σ σ σ
∂
∂∑
=0, = 1,
2 2
1 12 2
p q p qM I M Ip q p qS Sws j ws j
wj wjw j w j
F FX X
P Pσ σ σ σ
α αβ α α α
−∂ ∂ + + ∂ ∂
∑ ∑ (55a)
=0, = 1,
2 2
1 12 2
p q p qM I M Ip q p qS Sws j ws j
wj wjw j w j
F FX X
P Pσ σ σ σ
β βα β β β
+∂ ∂ + + ∂ ∂
∑ ∑ (55b)
=0, = 1,
1 1
1 12 2
p q p qM I M Ip q p qS Sb zs b zs
bz bzb z b z
F FX X
P Pσ σ σ σ
β ββ α β β
−∂ ∂ + + ∂ ∂
∑ ∑ (55c)
=0, = 1,
1 1
1 12 2
p q p qM I M Ip q p qS Sb zs b zs
bz bzb z b z
F FX X
P Pσ σ σ σ
α αα β α α
+∂ ∂ + + ∂ ∂
∑ ∑ (55d)
=0, = 2,1 1
2 2
p q p qM I M Ip q p qS Sb j b j
bj bjb j b j
F FX X
P Pσ σ σ σ
β α β αβ α β α
−∂ ∂ + + ∂ ∂
∑ ∑ (55e)
=0, = 2,1 1
2 2
p q p qM I M Ip q p qS Sb j b j
bj bjb j b j
F FX X
P Pσ σ σ σ
α β α βα β α β
+∂ ∂ + + ∂ ∂
∑ ∑ (55f)
=0, R ,
2 1 2 1
1 1 .2 2
p q p q p q p qM Ip q p q p q p q edun ISws zs ws zs
wz wzw z w z w z w z
F F F FX X
P P P Pσ σ σ σ σ σ σ σ
β α α β α α β β
∂ ∂ ∂ ∂ + + + − ∂ ∂ ∂ ∂ ∑ ∑ (55g)
３５
2.2 Disentangling different MS response
Using the newly defined separated excitation amplitudes in Tab. 2, one
can separate equations of motion (EOM) into EOMs with different MS
values. Of the nine types of the M ,R Ip qp q
X σ σ amplitudes shown in Tab. 1,
the last two amplitudes, M ,R IV CXα α
and M ,R IV CXβ β
, do not include the = 0SM
amplitudes and they are dropped in the EOM. For the remaining seven
types, the 1SM = ± contributions shown of the light gray background
in the last column of Tab. 1 and the = 2SM ± contributions (dark gray
background) are removed from the EOM of the = 0SM amplitudes as
described in the following.
Removal of the contributions with 0SM ≠ from the response equations
is illustrated by the example of the M ,
2
R Ius iX α amplitude matrix. The EOM
are first pre-multiplied by (1 ) 2i+ and then transformed according
to Eq. (55) to yield
=0, = 1,
2 2
12
M I M IS SI us i I us iX i Xα αω ω −+
=0,(0) (0)
2
1=2
M Iu i Su w ij j i uw ws j
wj w j
FF F X
Pβ α
β β α α αβ α
δ δ ∂ − + ∂ ∑
= 1, = 1,(0) (0)
2 2
M I M Iu i u iS Su w ij j i uw ws j ws j
wj w j w j
F Fi F F X X
P Pα α α α
α α α α α βα α β β
δ δ − + ∂ ∂ + − + + ∂ ∂ ∑ (56a)
=0, = 1, = 1,
1 1 1
12
M I M I M Iu i u i u iS S Sb zs b zs b zs
bz bzb z b z b z
F F FX i X X
P P Pβ α α α α α
β α ββ α α α β β
+ − ∂ ∂ ∂ + + + ∂ ∂ ∂ ∑ ∑ (56b)
３６
=0, = 2,(0) (0)1
2M I M Iu i u iS S
u b ij b j u b ij b jbj bjb j b j
F FF X F X
P Pβ α β α
β β β α β β β αβ α β α
δ δ − ∂ ∂ + + + + ∂ ∂ ∑ ∑ (56c)
=0,(0) (0) R ,
2 1 2 1
1 .2 2
M Iu i edun Iu i u iSz i uw ws zs z i uw ws zs
wz wzw z w z w z
F F FF X i F X
P P Pβ α α α α α
α α α αβ α α α β β
δ δ ∂ ∂ ∂ + − + + − + − ∂ ∂ ∂ ∑ ∑
(56d)
In Eqs. (56a) and (56b), the = 1SM ± amplitudes make purely
imaginary contributions into the response equations and are dropped
from the equations for the = 0SM amplitudes; the same is true for the
redundant part of the O→O transitions, see Eq. (56d). Although the
= 2SM − amplitude, Eq. (56c), is coupled with the = 0SM
contributions, its amplitude is neglected in the final response equation;
this is similar to the assumptions behind the standard SF-TDDFT
formalism. Hence, keeping only the = 0SM amplitudes the response
equations read
=0, =0, =0, =0, =0,
O C 11 O C 12 V O 13 V C 14 O O2 2 1 2 1
= ,M I M I M I M I M IS S S S SI s s s s sα α β β α
ω + + +X A X A X A X A X (57)
where amplitude vectors are =0,
O C2
( )M ISwjs α
X = =0,
2
M ISws jX α ,
=0,V O
1( )M IS
bzsβX =
=0,
1
M ISb zsX β ,
=0,V C( )M IS
bjβ αX =
=0,M ISb jX β α ,
=0,O O
2 1( )M IS
wzs sX =
=0,
2 1
M ISws zsX , and the coupling block matrices
are (0) (0)
11 ,( ) u iui wj u w ij j i uw
w j
FF F
Pβ α
β β α αβ α
δ δ∂
≡ − +∂
A , 12 ,( ) u iui bz
b z
FP
β α
β α
∂≡∂
A ,
(0)13 ,( ) u i
ui bj u b ijb j
FF
Pβ α
β ββ α
δ∂
≡ +∂
A and (0)
14 ,1( ) ( )2
u iui wz z i uw
w z
FF
Pβ α
α αβ α
δ∂
≡ − +∂
A .
It is noted that all the zeroth-order Fock matrices in Eq. (56) are
３７
originated from MS = +1 reference. Those from from MS = -1
reference will be represented with a tilde as (0)p qp qF σ σ
, and these satisfy
following equalities
(0) (0) (0) (0), .p q p q p q p qF F F Fα α β β β β α α= = (58)
The response equations for the MS = 0 amplitudes of the other six
types also can be obtained in a similar way, and collecting all these
equations can form a matrix equation as
=0,O C11 12 13 14
2
=0,V O21 22 23 24
1
=0,V C31 32 33 34
=0,O O41 42 43 44 45 46 47
2 1
54 55 56 57
64 65 66 67
74 75 76 77
M ISs
M ISs
M IS
MSs s
α
β
β α
XA A A A 0 0 0
XA A A A 0 0 0
XA A A A 0 0 0
XA A A A A A A
0 0 0 A A A A
0 0 0 A A A A
0 0 0 A A A A
=0,O C
2
=0,V O
1
=0,V C
=0,O O
2 1
=0, =0,V C V C
=0, =0,V O V O
1 1
=0, =0,O C O C
2 2
=
M ISs
M ISs
M IS
I M ISI s s
M I M IS S
M I M IS Ss s
M I M IS Ss s
α
β
β α
α β α β
α α
β β
ω
X
X
X
X
X X
X X
X X
,
(59)
where the amplitude vectors are =0, =0,
V C( ) =M I M IS Sbj b jX α βα β
X ,
=0, =0,V O 11
( ) =M I M IS Sbz b zss
X ααX , and
=0, =0,O C 22
( ) =M I M IS Swj ws js
X ββX , and 0 is the zero matrix
block. Complete specifications of the block matrices kmA ( , = 1 7k m )
are given in the Tab. 3.
３８
By spin symmetry, the block-coupling matrices from the = 1SM +
component are identical to the respective matrices from = 1SM − .
Hence, the matrix elements of A satisfy
(8 )(8 )= , , = 1,2,3.km k m k m− −A A (60)
In addition, A is a symmetric matrix ( = TA A ), see Tab. 3.
３９
( ) (0) (0)11 ,
u iu w ij j i uwui wj
w j
FF F
Pβ α
β β α αβ α
δ δ∂
≡ − +∂
A ( )12 ,u i
ui bzb z
FP
β α
β α
∂≡∂
A
( ) (0)13 , ,u i
u b ijui bjb j
FF
Pβ α
β ββ α
δ∂
≡ +∂
A ( ) (0)14 ,
12
u iz i uwui wz
w z
FF
Pβ α
α αβ α
δ ∂
≡ − + ∂ A
( )21 ,a v
av wjw j
FP
β α
β α
∂≡∂
A ( ) (0) (0)22 , ,a v
a b vz z v abav bzb z
FF F
Pβ α
β β α αβ α
δ δ∂
≡ − +∂
A
( ) (0)23 ,
a vj v abav bj
b j
FF
Pβ α
α αβ α
δ∂
≡ − +∂
A ( ) (0)24 ,
12
a va w vzav wz
w z
FF
Pβ α
β ββ α
δ ∂
≡ + ∂ A
( ) (0)31 ,
a ia w ijai wj
w j
FF
Pβ α
β ββ α
δ∂
≡ +∂
A ( ) (0)32 ,
a ij i abai bz
b z
FF
Pβ α
α αβ α
δ∂
≡ − +∂
A
( ) (0) (0)33 ,
a ia b ij j i abai bj
b j
FF F
Pβ α
β β α αβ α
δ δ∂
≡ − +∂
A ( )34 ,
12
a iai wz
w z
FP
β α
β α
∂≡
∂A
( ) (0)41 ,
12
u vj v uwuv wj
w j
FF
Pβ α
α αβ α
δ ∂
≡ − + ∂ A ( ) (0)
42 ,
12
u vu b vzuv bz
b z
FF
Pβ α
β ββ α
δ ∂
≡ + ∂ A
( )43 ,
12
u vuv bj
b j
FP
β α
β α
∂≡
∂A ( ) (0)
45 ,
12
u vj v uwuv wj
w j
FF
Pα β
β βα β
δ ∂
≡ − + ∂ A
( ) (0) (0) (0) (0)44 ,
1 12 2
u v u vu w vz z v uw u w vz z v uwuv wz
w z w z
F FF F F F
P Pβ α α β
β β α α α α β ββ α α β
δ δ δ δ ∂ ∂
≡ − + + − + ∂ ∂ A
( ) (0)46 ,
12
u vu b vzuv bz
b z
FF
Pα β
α αα β
δ ∂
≡ + ∂ A ( )47 ,
12
u vuv bj
b j
FP
α β
α β
∂≡
∂A
( )54 ,
12
a iai wz
w z
FP
α β
α β
∂≡
∂A ( ) (0) (0)
55 ,a i
a b ij j i abai bjb j
FF F
Pα β
α α β βα β
δ δ∂
≡ − +∂
A
( ) (0)56 ,
a iz i abai bz
b z
FF
Pα β
β βα β
δ∂
≡ − +∂
A ( ) (0)57 ,
a ia w ijai wj
w j
FF
Pα β
α αα β
δ∂
≡ +∂
A
( ) (0)64 ,
12
a va w vzav wz
w z
FF
Pα β
α αα β
δ ∂
≡ + ∂ A ( ) (0)
65 ,a v
j v abav bjb j
FF
Pα β
β βα β
δ∂
≡ − +∂
A
( ) (0) (0)66 ,
a va b vz z v abav bz
b z
FF F
Pα β
α α β βα β
δ δ∂
≡ − +∂
A ( )67 ,a v
av wjw j
FP
α β
α β
∂≡∂
A
( ) (0)74 ,
12
u iz i uwui wz
w z
FF
Pα β
β βα β
δ ∂
≡ − + ∂ A ( ) (0)
75 ,u i
u b ijui bjb j
FF
Pα β
α αα β
δ∂
≡ +∂
A
( )76 ,u i
ui bzb z
FP
α β
α β
∂≡∂
A ( ) (0) (0)77 ,
u iu w ij j i uwui wj
w j
FF F
Pα β
α α β βα β
δ δ∂
≡ − +∂
A
Table 3. Details of the block-coupling matrices. The zeroth-order Fock matrix
without a tilde denote that from MS = +1 reference, while that with tilde is
from MS = -1 reference. Explicit expression of the kernel, p q r sp q r sF Pσ σ σ σ∂ ∂ ,
with the collinear approximation is given in Eq. (38).
４０
2.3 Recovery of one-to-one relation between configuration and excitation amplitude
The excitation amplitudes =0,
2 1
M ISks msX ( , =k m O1,O2) correspond to the
type I configurations, see Fig. 2. Unlike the other six types, this type
of excitation amplitudes include amplitudes from both MS = +1 and -1
components of the mixed reference not from one. Figure 4 shows
configurations and the KS determinants, which are labeled as G
(ground), D (double), L (left), and R (right) according to the character
of the excitation. A subscript added to the label represents the MS
value of the parent component of the reference state. With these
notations, the amplitude =0,
O1 O22 1
M ISs sX corresponds to the KS determinants
1G± , =0,
2 12 1
M ISO s O sX to 1D± ,
=0,O1 O12 1
M ISs sX to 1L+ and 1R− , and
=0,O2 O22 1
M ISs sX to 1R+
and 1L− . Although =0,
O1 O22 1
M ISs sX and
=0,O2 O12 1
M ISs sX correspond to specific G
and D configurations, respectively, the signs of the respective
determinants originating from different components of the mixed
reference are opposite as shown in Fig. 4a and b, i.e., 1 1=G G+ −− ,
1 1=D D+ −− . On the other hand, =0,
O1 O12 1
M ISs sX corresponds to different
configurations L and R originating from = 1SM and 1− ,
respectively, see Fig. 5a. A similar correspondence of configurations
for =0,
O2 O22 1
M ISs sX is shown in Fig. 5b.
４１
Figure. 4. Type I configurations and notation of their Slater determinants
originating from spin flip transitions from the MS = +1 and -1 components of
mixed reference. The subscript at the notation denotes the MS value of the parent component.
４２
Figure. 5. Connection between one electron transition from the mixed-spin
reference and that from the open-shell configuration. Dotted curves show one-electron spin-flip excitation generating the given configuration.
４３
To obtain amplitudes corresponding to pure configurations G, D, L, and
R, the respective part of the orbital Hessian matrix A needs to be
modified in such a way as to resolve the sign changes (G, D) and mixed
configurations (L, R). Equivalently, the corresponding EOM could be
modified with the identical result. Before the modification, let us re-
write the affected blocks in the fourth row or column of the A matrix
in Eq. (59) in terms of the individual determinants shown in Fig. 4. The
elements 4 ( = 1,2,3)k kA in Eq. (59) correspond to the α β→
transitions from the = 1SM + component and the elements
4 ( = 5,6,7)m mA to the β α→ transitions from the = 1SM −
component. In terms of the determinants shown in Fig. 4, these
elements are represented as
( )4 1 1 1 1= , = 1,2,3,k kG kD kL kR k
+ + + +A A A A A (61)
( )4 1 1 1 1= , = 5,6,7.m mG mD mR mL m
− − − −A A A A A (62)
These block matrices satisfy the following symmetric relations as
(8 )1 1
(8 )1 1
(8 )1 1
(8 )1 1
=
= = 1,2,3,
=
=
kG k G
kD k D
kL k R
kR k L
k
−+ −
−+ −
−+ −
−+ −
A A
A A
A A
A A
(63)
The first and second terms of 44 O1 2, 1 2( ) O O OA (see Tab. 3) originate from
４４
the α β→ and β α→ transitions from the = 1SM + and = 1SM −
components, respectively. Hence, the individual contributions to this
element can be represented as 1 1G GA+ +
and 1 1G GA− −
, respectively. With
similar notations, the block matrix of the fourth row and column of the
A matrix can be written out as
44
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1
1 1 1
1=2
GG GD GL GR
DD DL DR
LL LR
RR
G G G G G D G D G L G R G R G L
D D D D D L D R D R D L
L L R R L R R L
R R L
A A A AA A A
A AA
A A A A A A A A
A A A A A A
A A A A
A A
+ + − − + + − − + + − − + + − −
+ + − − + + − − + + − −
+ + − − + + − −
+ + −
≡
+ + + +
+ + +
+ +
+
A
1L−
(64)
The lower triangular part is not shown in the above equations, because
the block matrix 44A is symmetric, 44 44= TA A . Parts of elements
satisfy the following symmetric relations as
1 1 1 1
1 1 1 1
1 1 1 1
1 1 1 1
=
= ( , ) = ( , ), ( , ), ( , ), ( , ) ).
=
=
l G m G
l D m D
l L m R
l R m L
A A
A Al m G G D D L R R L
A A
A A
+ + − −
+ + − −
+ + − −
+ + − −
(65)
where the symmetry between 1L+ and 1R− and between 1R+ and 1L−
is seen.
４５
The Hessian A matrix can be written out as
11 12 13 1 1 1 11 1 1 1
22 23 2 2 2 21 1 1 1
33 3 3 3 31 1 1 1
3 2 11 1 1
3 2 11 1 1
3 2 11 1 1
3 2 11 1 1
33 23 13
22 12
11
=
G D L R
G D L R
G D L R
GG GD GL GR G G G
DD DL DR D D D
LL LR R R R
RR L L L
A A A A
A A A
A A
A
+ + + +
+ + + +
+ + + +
− − −
− − −
− − −
− − −
A A A A A A A 0 0 0
A A A A A A 0 0 0
A A A A A 0 0 0
A A A
A A AAA A A
A A A
A A AA A
A
,
(66)
where symmetry relations of Eq. (60) were used at ( , = 5,6,7)km k mA
and the lower triangular part is not shown with the same reason of Eq.
(64). The advantage of the new labeling scheme for the matrix
elements of A is that the contributions from = 1SM + and = 1SM −
components can now be clearly identified.
With the new notation, amplitudes corresponding to G , D , L and R
of Fig. 4 can be easily obtained by exchanging 1 1R L− −↔ in all the
elements of Eqs. (64), (65) and (66) with simultaneous sign change of
the elements for 1G− and 1D− ; double replacement leaves the sign
unchanged. Such an exchange is possible, because, ideally, amplitude
and configuration should have one-to-one correspondence. For an
４６
example, let us consider matrix elements of GLA and GRA , i.e.,
1 1 1 1G L G RA A+ + − −
+ and 1 1 1 1G R G LA A+ + − −
+ , respectively. As discussed above, one
makes a sign change of 1 1G LA− −
and 1 1G RA− −
terms connected with 1G− ,
and exchanges these terms to bring 1L+ together with 1L− and 1R+
together with 1R− . Then, the modified elements become 1 1 1 1G L G LA A+ + − −
−
and 1 1 1 1G R G RA A+ + − −
− , which are denoted as GLB and GRB , in the following.
The Hessian of Eq. (66) becomes modified by applying these
discussions. The resulting Hessian matrix (0)sA is given by
11 12 13 1 1 1 11 1 1 1
22 23 2 2 2 21 1 1 1
33 3 3 3 31 1 1 1
3 2 11 1 1
(0) 3 2 11 1 1
3 2 11 1 1
3 2 11 1 1
33 23 13
2
=
G D L R
G D L R
G D L R
GG GD GL GR G G G
DD DL DR D D Ds
LL LR L L L
RR R R R
B B B B
B B B
B B
B
+ + + +
+ + + +
+ + + +
− − −
− − −
− − −
− − −
− − −
− − −
A A A A A A A 0 0 0
A A A A A A 0 0 0
A A A A A 0 0 0
A A A
A A AAA A A
A A A
A A AA 2 12
11
,
AA
(67)
where the superscript (0) in the orbital Hessian matrix do not denote
the zeroth-order quantity but will denote Hessian without spin-pairing
coupling, and rearranged coupling between type I configurations are
４７
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1
1 1 1 1
1=2
GG GD GL GR
DD DL DR
LL LR
RR
G G G G G D G D G L G L G R G R
D D D D D L D L D R D R
L L L L L R L R
R R R R
B B B BB B B
B BB
A A A A A A A A
A A A A A A
A A A A
A A
+ + − − + + − − + + − − + + − −
+ + − − + + − − + + − −
+ + − − + + − −
+ + − −
+ + − −
+ − −
+ +
+
.
(68)
As 1 1 1 1
=G L G RA A+ + − −
and 1 1 1 1
=G R G LA A+ + − −
in Eq. (65), a new relation GLB =
GRB− holds. Likewise, new symmetric relations of DLB = DRB− and LLB =
RRB also appear. These symmetric relations completely eliminate the
spin-contamination of type I in the response states.
In addition, with spin-pairing coupling which will be introduced in
subsection 3 of Spin-pairing coupling, Hessian matrix sA is given by
11 12 13 1 1 1 1 13 12 111 1 1 1
22 23 2 2 2 2 23 22 211 1 1 1
33 3 3 3 3 33 32 311 1 1 1
3 2 11 1 1
3 2 11 1 1
3 2 11 1 1
3 21 1
=
G D L R
G D L R
G D L R
GG GD GL GR G G G
DD DL DR D D Ds
LL LR L L L
RR R R
B B B B
B B B
B B
B
+ + + +
+ + + +
+ + + +
− − −
− − −
− − −
− −
− − −
− − −
A A A A A A A C C C
A A A A A A C C C
A A A A A C C C
A A A
A A AAA A A
A A A 11
33 23 13
22 12
11
.
R−
A A AA A
A
(69)
４８
2.4 Separating matrix equations for singlet and triplet response states
Pure spin configurations are obtained by pairing the respective
configurations originating from MS = +1 and -1 references for the type
I, II and III (see Fig. 2), which is achieved by a unitary transformation
O O
O O
V V
V V
O O
O O
2 0 0 01 0 2 0 0' = ,2 0 0 1 1
0 0 1 1
C C
V V
C C
C C
V V
C C
− − − −
I 0 0 0 0 0 0 0 0 I0 I 0 0 0 0 0 0 I 00 0 I 0 0 0 0 I 0 0
0 0 0 0 0 0
0 0 0 0 0 0U0 0 0 0 0 00 0 0 0 0 00 0 I 0 0 0 0 I 0 00 I 0 0 0 0 0 0 I 0
I 0 0 0 0 0 0 0 0 I
(70)
with ( )O , =C uw ijui wj δ δI , ( )O , =V ab vzav bz δ δI , ( )V , =C ab ijai bj δ δI and ( )O , =O uw vzuv wz δ δI .
The rotated amplitude vector by the transformation matrix is given by
=0, =0,O C O C
2 2=0, =0,
V O V O1 1
=0, =0,V C V C
=0,G
=0,D
=0, =0,L R
=0, =0,L R
=0, =0,V C V C
=0, =0,V O V O
1 1
( ) / 2
( ) / 2
( ) / 2
' =( ) / 2
( ) / 2
( ) / 2
(
M I M IS Ss s
M I M IS Ss s
M I M IS S
M IS
M IS
I M I M IS S
M I M IS S
M I M IS S
M I MS Ss s
X
X
X X
X X
α β
β α
β α α β
β α α β
β α
+
+
+
+
−
−
−
X X
X X
X X
U X
X X
X X
=0, =0,O C O C
2 2
.
) / 2
( ) / 2
I
M I M IS Ss sα β
−
X X
(71)
４９
and the orbital Hessian matrix (0)sA of Eq. (67) can be split into two
blocks with the rotation as
11 12 13 1 11 1
21 22 23 2 21 1( )(0)
31 32 33 3 31 1
1 1 2 2 3 31 1 1 1 1 1
= ,
L R
L R
TL R
L R L R L R LL LRB B
+ +
+ +
+ +
+ + + + + +
+
+
+ + + + +
A A A A A
A A A A A
A A A A A A
A A A A A A
(72)
3 2 11 1 1
3 2 11 1 1
3 3 2 2 1 11 1 1 1 1 1( )(0)
3 3 3 3 33 23 131 1 1 1
2 2 2 2 23 22 121 1 1 1
1 1 1 11 1 1 1
2 2 2 2
2 2 2 2
2 2
= 2 2
2 2
2 2
GG GD GL G G G
GD DD DL D D D
GL DL LL LR L R L R L R
SG D L R
G D L R
G D L R
B B B
B B B
B B B B
+ + +
+ + +
+ + + + + +
+ + + +
+ + + +
+ + + +
− − − −
−
−
−
A A A
A A A
A A A A A A
A A A A A A A A
A A A A A A A
A A A A A13 12 11
.
A A
(73)
Hence, two matrix equations
( )(0) ( )(0) ( )(0)( )(0)= ,T T I T
I T IΩA X X (74)
( )(0) ( )(0) ( )(0)( )(0)= ,S S I S
I S IΩA X X (75)
are obtained, for the triplet (T) and singlet (S) state amplitudes. In
these equations, the excitation amplitude vectors are
=0, =0,( )(0),CO O C O C2 2
=0, =0,( )(0),OV V O V O( )(0) 1 1
=0, =0,( )(0),CV V C V C
=0, =0,( )(0),OOT L R
( ) / 2
( ) / 2,
( ) / 2
( ) / 2
M I M IT I S Ss s
M I M IT I S Ss sT
IM I M IT I S S
M I M IT I S SX X X
α β
β α
β α α β
+
+ ≡ =
+ +
X X X
X X XX
X X X (76)
５０
=0,( )(0),GG
=0,( )(0),DD
=0, =0,( )(0),L ROOS( )(0)
=0, =0,( )(0),V C V CCV
=0, =0,( )(0),V O V OOV 1 1
( )(0),OCO
( ) / 2
( ) / 2
( ) / 2
(
M ISS I
M ISS I
M I M IS SS I
SI M I M IS SS I
M I M IS SS Is s
S Is
XX
XX
X XX
β α α β
β α
− ≡ = − −
XX XX
X XX
XX=0, =0,C O C
2 2
.
) / 2M I M IS Ssα β
−
X
(77)
Here also, the superscript (0) do not denote the zeroth-order quantity
but will denote quantity without spin-pairing coupling. The new
response equations (74) and (75) yield spin-adapted excited states,
where a complete decontamination of the type I, II and type III
configurations is achieved. Hence, clean separation of triplet and
singlet states is achieved in MRSF-TDDFT; which is an advantage
before the standard SF-TDDFT formalism.
For the type IV configurations, only one missing configuration (out of
five) is recovered in MRSF-TDDFT and spin-adaptation of this type
of configurations remains incomplete. However, contribution of these
configurations into the low lying excited states is expected to be small
and the resulting spin contamination insignificant.
Likewise, rotation of the Hessian matrix sA of Eq. (69), which can be
split into two blocks
５１
11 11 12 12 13 13 1 11 1
12 12 22 22 23 23 2 21 1( )
13 13 23 23 33 33 3 31 1
1 1 2 2 3 31 1 1 1 1 1
= ,
L R
L R
TL R
L R L R L R LL LRB B
+ +
+ +
+ +
+ + + + + +
+ + + +
+ + + +
+ + + + + + + +
A C A C A C A A
A C A C A C A A
A A C A C A C A A
A A A A A A
(78)
3 2 11 1 1
3 2 11 1 1
3 3 2 2 1 11 1 1 1 1 1( )
3 3 3 3 33 33 23 23 13 131 1 1 1
2 2 2 2 23 23 22 22 12 121 1 1 1
1
2 2 2 2
2 2 2 2
2 2
= 2 2
2 2
2
GG GD GL G G G
GD DD DL D D D
GL DL LL LR L R L R L R
SG D L R
G D L R
B B B
B B B
B B B B
+ + +
+ + +
+ + + + + +
+ + + +
+ + + +
− − − −
− − − −
− − − −
A A A
A A A
A A A A A A
A A A A A A C A C A C
A A A A A C A C A C
A 1 1 1 13 13 12 12 11 111 1 1 1
,
2G D L R+ + + +
− − − −
A A A A C A C A C
(79)
leads to two sets of response equations
( ) ( ) ( )( )= ,T T I T
I T IΩA X X (80)
( ) ( ) ( )( )= ,S S I S
I S IΩA X X (81)
2.5 Expectation value of S2 operator for response states of MRSF-TDDFT
It has been a common practice that the spin-contamination of TDDFT
is measured by the operation of 2S on a wave function of
noninteracting system. The 2S operator is represented as
2 = ( 1) ,z zS S S S− ++ +S with (82)
５２
= ( ),z z
mS s m∑
(83)
= ( ),
mS s m± ±∑
(84)
where ( )zs m , ( )s m+ , and ( )s m− are the one-electron operators of the
spin z -component, spin raising and lowering operators for the m th
electron, respectively. In the second quantization, the S± operators
are
†= ,p pp
S a aβ α− ∑ (85)
†= ,q qq
S a aα β+ ∑ (86)
where the p and q indices are running over all the MOs and the
†p p
a σ , p pa σ ( = ,pσ α β ) are the creation and annihilation operators of
electron in the pp
σφ orbital. Therefore, the last term of Eq. (82) can be
written as
† †
,= .p p q q
p qS S a a a aβ α α β− + ∑ (87)
With the help of the anti-commutation relations, the above equation is
re-written as
† † †
,= .p p p q p q
p p qS S a a a a a aβ β β α α β− + −∑ ∑ (88)
The second term on the right hand side of above equation can be split
as
† † † † †
,= .p p p p p p p q p q
p qp pp q
S S a a a a a a a a a aβ β β α α β β α α β− +
≠
− −∑ ∑ ∑ (89)
５３
The three terms on right hand side of Eq. (89) correspond to the
operators of the number of electrons in the β -spin MOs, in the doubly
occupied MOs, and in the open-shell configurations with two singly
occupied MOs, respectively. The MR-SF-TDDFT wavefunction for an
excited state with = 0SM is
=0, = 1 =0, = 1 =0, = 1 =0, = 1( )
O1 O2 O2 O1 O1 O1 O2 O2= M I M M I M M I M M I MS S S S S S S S SG D L RX X X Xβ α β α β α β α
+ + + +Ψ Ψ + Ψ + Ψ + Ψ
( ) ( )=0, = 1 =0, = 1 =0, = 1 =0, = 1
2 2 1 1
M I M M I M M I M M I MS S S S S S S Sus i u i us i u i a ws a w a ws a w
ui awX X X Xα β α β α β β β α α α β
+ − + −+ Ψ + Ψ + Ψ + Ψ∑ ∑
( )=0, = 1 =0, = 1M I M M I MS S S S
a i a i a i a iai
X Xβ α β α α β α β+ −+ Ψ + Ψ∑
(90)
where = 1MS
p qβ α+Ψ and
= 1MSp qα β
−Ψ are the determinants obtained by the spin-
flip transitions from the = 1SM + and = 1SM − components of the
mixed reference state, respectively. The first four terms on the right
hand side of Eq. (90) correspond to the type I configurations, see Fig.
2, while 5th, 6th, 7th terms represent the type II, III, IV, respectively.
Using IΨ the expectation value of 2S becomes
2 = ( 1) .I I I z z I I IS S S S− +Ψ Ψ Ψ + Ψ + Ψ ΨS (91)
where the first term on the right hand side vanishes, since = 0SM , and
the second term is given by
( ) ( )2 2=0, =0, =0, =0,= 1 2M I M I M I M IS S S SI I G D L RS S X X X X− +Ψ Ψ − − +
=0, =0, =0, =0, =0, =0,
2 2 1 12( ).M I M I M I M I M I M IS S S S S S
a i a ius i us i a ws a wsus aw ai
X X X X X Xβ α α βα β β α+ + +∑ ∑ ∑ (92)
５４
Magnitudes of two amplitudes for each pair which is =0, =0,( , )M I M IS S
L RX X ,
=0, =0,
2 2( , )M I M IS S
Os C Os CX Xα β , =0, =0,
1 1( , )M I M IS S
V Os V OsX Xβ α , or =0, =0,( , )M I M IS S
V C V CX Xβ α α β are same, while
those sign are different and same for singlet and triplet response
states, respectively. Thus, with the orthonormal condition, the
expectation values of 2S for singlet and triplet response states are
always 0 and 2, respectively. This shows that MRSF-TDDFT
eliminates spin contamination of SF-TDDFT for singlet and triplet
response states. As discussed in the beginning of this section, there
is still quintet mixing for C→V configurations but it is minor
contribution for low-lying excited states.
2.6 Dimensional transformation matrix
The different singlet and triplet response dimensions of MRSF-TDDFT
as compared to SF-TDDFT could introduce complications to the
subsequent derivations and potentially require a major modification to
the existing SF-TDDFT code. Therefore, in this subsection, we
introduce dimensional-transformation ( )SpqU and ( )T
pqU matrices, which
cause the singlet and triplet response dimensions of MRSF-TDDFT to
be equal to that of SF-TDDFT. With these transformation matrices,
５５
the expanded X vectors, which have the same dimension of SF-TDDFT,
can be represented as ( ) ( )(0)S Spq pqU X and ( ) ( )(0)T T
pq pqU X for the singlet and
triplet response spaces, respectively. For example, the expanded X
vectors of G and D configurations for the singlet and triplet spaces,
respectively, are defined as:
( ) ( )(0) ( )(0)2 1 ,S G S S
pq G pO qO GU X Xδ δ≡ (93a)
( ) ( )(0) ( )(0)1 2 ,S D S S
pq D pO qO DU X Xδ δ≡ (93b)
( ) ( )(0) 0,T G Tpq GU X ≡ (94a)
( ) ( )(0) 0.T D Tpq DU X ≡ (94b)
Likewise, the expanded X vectors of OOS and OOT configurations
for the singlet and triplet spaces, respectively, are defined as:
( )( )(0) ( )(0)1 1 2 2
1 ,2
OOS S Spq OOS pO qO pO qO OOSU X Xδ δ δ δ≡ − (95)
( )( )(0) ( )(0)1 1 2 2
1 .2
OOT T Tpq OOT pO qO pO qO OOTU X Xδ δ δ δ≡ + (96)
Note that the one-dimensional excitation amplitudes of ( )(0)SOOSX and
( )(0)TOOTX are represented as two-dimensional excitation amplitudes of
( ( ) ( )(0)1 1S OO S
O O OOSU X , ( ) ( )(0)2 2S OO S
O O OOSU X ) and ( ( ) ( )(0)1 1T OO T
O O OOTU X , ( ) ( )(0)2 2
T OO TO O OOTU X ), respectively. As
compared to the OO type, the COmpqU , OmV
pqU , and CVpqU are defined
without changing their dimensions as:
( )(0) ( )(0)< 1 ,COm k k
pq pq p O qOm pqU X Xδ δ≡ (97)
５６
( )(0) ( )(0)> 2 ,OmV k k
pq pq pOm q O pqU X Xδ δ≡ (98)
( )(0) ( )(0)< 1 > 2 ,CV k k
pq pq p O q O pqU X Xδ δ≡ (99)
where = 1, 2m , = ,k S T , and
< 1 < 1= 1 f < 1, = 0 f 1,p O p Oor p O or p Oδ δ ≥ (100a)
> 2 > 2= 1 f > 2, = 0 f 2.q O q Oor q O or q Oδ δ ≤ (100b)
As a result, the expanded dimensions of both singlet and triplet
response spaces are equal to occ virn nα β of SF-TDDFT. With the help of
these transformation matrices, the expanded single and triplet
response spaces can be elegantly defined, respectively, as:
( ) ( ) ( ) 1 2 1 2 ,S S G S D OOS CO CO O V O V CVpq pq pq pq pq pq pq pq pqU U U U U U U U U≡ + + + + + + + (101a)
( ) ( ) ( ) 1 2 1 2 .T T G T D OOT CO CO O V O V CVpq pq pq pq pq pq pq pq pqU U U U U U U U U≡ + + + + + + + (101b)
With the collinear approximation, the singlet and triplet orbital
Hessians of Eqs. (72) and (73) can be succinctly represented with a
single form of:
( ) ( )(0) ( ) ( ), = ,k k k
pq rs pq pr qs qs pr H rsA U F F c pr sq Uβ αδ δ− − (102)
where = ,k S T and the zeroth-order Fock matrix from MS = +1 is
represented with more concise notation as (0)p qpq p qF Fσ σ σ= and all Fock
matrices from MS = -1, (0)p qp qF σ σ
, is converted to that from MS = +1. The
singlet and triplet response equations can be represented as:
５７
( )(0) ( )(0) ( )(0), ( )(0)= ,k k k
pq rs rs k pqA X XΩ (103)
the ( )(0)SΩ and ( )(0)TΩ are their eigenvalues or excitation energies from
the reference.
3. Spin-pairing coupling
Because no coupling occurs between the responses originating from
the two different references of the MR-RDM, a posteriori coupling was
introduced as
= 1 = 1,
ˆ= ,M MS Spq rs SP p q r sA c Hα β β α
+ −′ Ψ Ψ (104)
where = 1MSp qα β
+Ψ and = 1MSr sβ α
−Ψ are the bra and ket vectors for
configurations originating from the = 1SM + and = 1SM − components
of the mixed reference state, respectively. The pairing-strength
coefficient, SPc , is adjustable depending on situations or a specific
molecule. The best pairing strength may be able to be determined by
benchmarking calculations. However, by default one can use the
pairing strength with the same value of HF exchange mixing
coefficient, i.e., SP HFc c= . After defining a sign function for the singlet
and triplet states as
s ( ) = 1, if = ,gn k k S+ (105a)
s ( ) = 1, if = ,gn k k T− (105b)
５８
the two types of couplings can be defined by:
( )( )intra, s ( ) ,k
pq rs HH gn k c ps rq≡ (106a)
( ) ( ) ( )inter, s ( ) ,k
pq rs HH gn k c pq rs pr sq≡ − (106b)
where = ,k S T . With these, the spin-pairing coupling in Eq. (104) for
the singlet and triplet response equations in Eq. (74) and (75),
respectively, are represented as:
( ) ( )intra 1 2 1 2 ( )intra 1 2 1 2, , ,= ( )( ) ( )( )k k CO CO CO CO k O V O V O V O V
pq rs pq pq rs rs pq pq rs rspq r s pq rsA H U U U U H U U U U′ − − + − −
( )inter 1 2 2 1 1 2 2 1, ( )k CO O V CO O V O V CO O V CO
pq rs pq rs pq rs pq rs pq rsH U U U U U U U U+ + + + (107)
Note that the underline notation used in the indices of the coupling
( )intra,
kpq rsH in Eq. (107) is defined as:
2, if = 1,p O p O≡ (108a)
1, if = 2.p O p O≡ (108b)
As a result, the orbital Hessians for the singlet and triplet responses
with the spin-pairing coupling can be simply given by:
( ) ( )(0) ( ), , ,= .k k k
pq rs pq rs pq rsA A A′+ (109)
It should be noted that even with the couplings, the excitation
amplitudes of singlet ( ( )SpqX ) and triplet configurations ( ( )T
pqX ) are
completely decoupled from each other. In addition, the response
spaces can be expanded with the same dimensional-transformation
５９
( )SpqU and ( )T
pqU matrices in Eq. (101a) and (101b). Finally, the singlet
and triplet response equations with the spin-pairing coupling are given
by:
( ) ( ) ( ), ( )= ,k k k
pq rs pq k pqA X XΩ (110)
where = ,k S T and the ( )SΩ and ( )TΩ are their eigenvalues or
excitation energies.
６０
ANALYTIC ENERGY GRADIENT OF MIXED-
REFERENCE SPIN-FLIP TDDFT
An analytic gradient with respect to nuclear coordinates represents
the essential quantity for the vast majority of quantum mechanical
applications such as geometry optimization, reaction path following,
and molecular dynamics simulations. In addition, gradients of excited
states are crucial in the emerging field of nonadiabatic dynamics. The
analytic energy gradient of LR- and SF-TDDFT is formulated by using
the Lagrangian formalism.[46, 49] Likewise, that of MRSF-TDDFT
can be obtained with similar way[50] which will be describing in this
section. In the first subsection of Lagrangian, the Lagrangians of both
singlet and triplet response states are defined with two new Lagrange
multipliers W and Z. These two multipliers can be determined by
orbital stationary condition described by subsections 2 and 3,
respectively. With the determined multipliers satisfying orbital
stationary conditions, analytic energy gradient can be represented by
concise form which will be derived in the last subsection 4.
６１
1. Lagrangian
The reference of MRSF-TDDFT is optimized by variational conditions
for the restricted open-shell Kohn-Sham MOs in the Guest-Saunders
canonicalization.[51]
= ,pq pq pqF F Fα β+ (111)
with
/ 2 0 / 2= 0 / 2 ,
/ 2 / 2
ij ib
pq xy xb
aj ay ab
F FF F F
F F F
α α
α α α
α α α
(112)
/ 2 / 2= / 2 0 .
/ 2 0 / 2
ij iy ib
pq xj xy
aj ab
F F FF F F
F F
β β β
β β β
β β
(113)
The off-diagonal blocks represent the variational conditions, i.e.,
rotations between C-O, C-V, and O-V orbitals that go to zero.
MRSF-TDDFT yields two independent Lagrangian for singlet ( =k S )
and triplet response states ( =k T ) as:
( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( )[ , , , , ] = [ , ] 2k k k k k k kk k ia ia ix ix
ia ixL G Z F Z Fσ σ σ σ
σ σ
Ω Ω + +∑ ∑X C Z W X
( ) ( )2 ( ),k kxa xa pq pq pq
xa p qZ F W Sσ σ σ σ
σ σ
δ≤
+ − −∑ ∑∑ (114)
where the vector C and pqS σ are the MO coefficients and MO overlap
integral, respectively, and the ( )kZ and ( )kW vectors are
undetermined Lagrange multipliers. The first term on the right-hand
６２
side is:
, , , ,
( ) ( ) ( ) ( ) ( ) ( )( ) , ( )[ , ] = ( 1).
C OO V C OO Vk k k k k k
k pq pq rs rs k pq pqpr qs p q
G X A X X XΩ −Ω −∑∑ ∑∑X (115)
It is noteworthy that only the ( )( )[ , ]kkG ΩX term differs in SF- and
MRSF-TDDFT except for the undetermined Lagrange-multipliers ( )kZ
and ( )kW .
2. Orbital stationary condition I (Coupled perturbed
Hartree-Fock equation): Lagrange multiplizer Z
Two independent sets of orbital stationary conditions for singlet ( =k S )
and triplet response states ( =k T ) are defined, respectively, as
( ) ( )
= 0.k k
u ut t
L Lc cc cµ α µ β
µ µµ α µ β
∂ ∂+
∂ ∂∑ ∑ (116)
From this condition, the following ( )kZ -vector equation can be derived
as
, ,
( ) ( ), = , , , ,
C OO Vk k
pq rs rs pqr s
J Z R p C O q O V− ∈ ∈∑∑ (117)
where the unique spin-independent ( )kZ vector (with the bar symbol)
is introduced as:
( ) ( )= ,k kix ixZ Z β (118a)
( ) ( )= ,k kxa xaZ Z α (118b)
６３
( ) ( ) ( )= = , otherwise, 0.k k kia ia iaZ Z Zα β (118c)
The orbital Hessian ,pq rsJ of MRSF-TDDFT takes the identical forms
of SF-TDDFT[46] as:
( ) ( ) ( ), ,1 1= [ ] ,
2 2 2xcH
ix jy ix jy ij xy xy ijcJ ix jy iy jx ij xy f F Fβ β β βδ δ− + + − + (119a)
( ) ( ) ( ), , ,1= 2 [ ] ,
2 2xc xcH
ia jy ia jy ia jy ya ijc
J ia jy iy ja ij ya f f Fα β β β β δ− + + + + (119b)
( ), ,1= ,2
xcxa jy xa jy ja xyJ xa jy f Fα β αδ+ −
(119c)
( ) ( ) ( ), , , , ,= 4 [ ] ( ) ,xc xc xc xcia jb H ia jb ia jb ia jb ia jb a i ij abJ ia jb c ib ja ij ab f f f fα α α β β α β β δ δ− + + + + + + −
(119d)
( ) ( ) ( ), , ,1= 2 [ ] ,2
xc xcxa jb H xa jb xa jb jx abJ xa jb c xb ja jx ab f f Fα α α β αδ− + + + − (119e)
( ) ( ) ( ), ,1 1 1= [ ] ,
2 2xc
xa yb xa yb xy ab ab xyH
J xa yb xb ya xy ab f F Fc α α α αδ δ− + + − + (119f)
where ,xc
pq rsf σ τ represents the matrix elements of the second functional
derivatives of the exchange-correlation functional with respect to the
electron density. The spin-state-specific ( )kpqR on the right-hand side
of Eq. (7) for the singlet ( =k S ) and triplet states ( =k T ) are given by:
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )1= [ ] [ , ] [ , ] [ , ],2
k k k k k k k kix ix ix xi xiR H H H Hβ α α β
+ + − −T X X X X X X (120a)
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )1= [ ] [ , ] [ , ] [ , ],2
k k k k k k k kxa xa xa xa axR H H H Hα α β β
+ + + −T X X X X X X (120b)
６４
( ) ( ) ( ) ( ) ( ) ( ) ( )1= ( [ ] [ ]) [ , ] [ , ],2
k k k k k k kia ia ia ia aiR H H H Hα β α β
+ ++ + −T T X X X X (120c)
where
( ) ( ) ( ),[ ] 2 2 [ ] ,xcpq pq rs H rs
rsH V pq rs f c ps rq pr sq Vσ σ τ στ τ
τ
δ+ ≡ + − +∑ (121)
with
,
( ) ( ) ( ) ( ) ( ) , , , ,O V
k k k k kpr pq pq rq rq
qT U X U X p r C Oα ≡ − ∈∑ (122a)
,
( ) ( ) ( ) ( ) ( ) , , , .C O
k k k k kqs pq pq ps ps
pT U X U X q s O Vβ ≡ ∈∑ (122b)
And the ( ) ( )[ , ]k ktuH σ X X of Eq. (120) is defined as
( ) ( ) (0) ( ) ( ) i ( ) ( ) i ( ) ( )[ , ] [ , ] [ , ] [ , ]k k k k ntra k k ntra k ktu tu tu Cx Cy tu xV yV
xy xyH H H Hσ σ σ σ≡ + +∑ ∑X X X X X X X X
i ( ) ( ) i ( ) ( )[ , ] [ , ],nter k k nter k ktu Cx xV tu xV C x
x xH Hσ σ+ +∑ ∑X X X X (123)
where
( ), ,
(0) ( ) ( ) ( ) ( ) ( ) ( )[ , ] ,C OO V
k k k k k ktu tq tq ur qs qs ur H rs rs
r qsH U X F F c ur sq U Xα β αδ δ≡ − −∑∑X X (124a)
( ), ,
(0) ( ) ( ) ( ) ( ) ( ) ( )[ , ] ,C OO V
k k k k k ktu pt pt pr us us pr H rs rs
pr sH U X F F c pr su U Xβ β αδ δ≡ − −∑∑X X (124b)
and
1i ( ) ( ) ( ) ( )i ( )
,[ , ] ( 1) ,C O
ntra k k Cx k k ntra Cy kxytu Cx Cy tq tq uq r s rs rs
r qsH U X H U X
δ
α
−≡ − ∑∑X X (125a)
1i ( ) ( ) ( ) ( )i ( )
,[ , ] ( 1) ,O V
ntra k k xV k k ntra yV kxytu xV yV tq tq uq rs rs rs
r qsH U X H U X
δ
α
−≡ − ∑∑X X (125b)
i ( ) ( ) ( ) ( )i ( ),[ , ] ,
O Vnter k k Cx k k nter yV k
tu Cx yV tq tq uq rs rs rsrq s
H U X H U Xα ≡ ∑∑X X (125c)
６５
i ( ) ( ) ( ) ( )i ( ),[ , ] ,
C O Vnter k k xV k k nter Cy k
tu xV Cy tq tq uq rs rs rsr s q
H U X H U Xα ≡ ∑∑∑X X (125d)
1i ( ) ( ) ( ) ( )i ( )
,[ , ] ( 1) ,C O
ntra k k Cx k k ntra Cy kxytu Cx Cy pt pt pu r s rs rs
pr sH U X H U X
δ
β
−≡ − ∑∑X X (125e)
1i ( ) ( ) ( ) ( )i ( )
,[ , ] ( 1) ,O V
ntra k k xV k k ntra yV kxytu xV yV pt pt pu rs rs rs
pr sH U X H U X
δ
β
−≡ − ∑∑X X (125f)
i ( ) ( ) ( ) ( )i ( ),[ , ] ,
C O Vnter k k Cx k k nter yV k
tu Cx yV pt pt pu rs rs rsp r s
H U X H U Xβ ≡ ∑∑∑X X (125g)
i ( ) ( ) ( ) ( )i ( ),[ , ] .
C Onter k k xV k k nter Cy k
tu xV Cy pt pt pu rs rs rsr pr
H U X H U Xβ ≡ ∑∑X X (125h)
Four terms on the right-hand side of Eq. (123) except for the first
term are derived from the spin-pairing coupling in Eq. (107). Without
these terms, all equations for the ( )kZ -vector equation are almost
same as those of SF-TDDFT.[46] Only the difference is using the
expanded excitation amplitudes, ( ) ( )k kpq pqU X , in MRSF-TDDFT. This is
great advantage since one can simply utilize the same existing routines
for SF-TDDFT.
3. Orbital stationary condition II: Lagrange multiplizer W
After ( )kZ vector is determined by solving Eq. (117), the relaxed
difference density matrix ( )kP can thus be calculated as:
( ) ( ) ( )= .k k kpq pq pqP T Zσ σ σ+ (126)
６６
If we define the spin-state-specific ( )kpqW as
( ) ( ) ( ) ,k k kpq pq pqW W Wα β≡ + (127)
the other Lagrange multiplier ( )kpqW can be obtained by:
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )1= [ , ] [ , ] [ , ] [ ]2
k k k k k k k kix xi xi xi ixW H H F Hα β α α
++ + +X X X X X X P
( ) ( )1 1 ,2 2
k kij jx ia xa
j aF Z F Zβ α+ +∑ ∑ (128a)
( ) ( ) ( ) ( ) ( )1= [ , ] ,2
k k k k kia ai i ia ix xa
xW H Z F Zβ α+ + ∑X X (128b)
( ) ( ) ( ) ( ) ( ) ( ) ( )1 1= [ , ] [ , ] ,2 2
k k k k k k kxa ax ax xy ya ix ia
y iW H F F Z F Zβ β α α+ + +∑ ∑X X X X (128c)
( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ( )11 = [ , ] [ , ] [ ] [ ] ,2
k k k k k k kij ij ij ij ij ijW H F H H i jα α α βδ + ++ + + + ≤X X X X P P
(128d)
( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )1 = [ , ] [ , ] [ , ] [ , ]k k k k k k k k kxy xy xy xy ij ijW H H F Fα β α βδ+ + + +X X X X X X X X
( )1 [ ],2
kxyH x yα++ ≤P (128e)
( )( ) ( ) ( ) ( ) ( )1 = [ , ] [ , ], ,k k k k kab ab ab abW H F a bα αδ+ + ≤X X X X (128f)
where
( ) ( ) ( ) ( )[ , ] = ,k k k ktu tq qs usF X F Xα β−X X (129a)
( ) ( ) ( ) ( )[ , ] = .k k k ktu pt pr ruF X F Xβ αX X (129b)
Also in this case, form of equations are exactly same as those of SF-
TDDFT[46] without four terms originated from spin-pairing coupling
in Eq. (107).
６７
4. Analytic energy gradient with respect to the nuclear
coordinate
From the stationary condition of Lagrangian for a nuclear coordinate
(ξ ) of
( )
= 0,kLξ
∂∂
(130)
the analytic gradient of the excitation energy ( ( )kξΩ ) can be obtained
by:
)( ) ( ) ( )( ) ,
,= ( | ,k k k
k h P S W ξξ ξ ξµν µνσ µν µνσ µνσ κλτ
µνσ µνσ µνσ κλτ
µν κλΩ − + Γ∑ ∑ ∑ (131)
where the superscript ξ denote the derivative with respect to the
nuclear coordinate. hξµν and )( | ξµν κλ are the derivatives of one- and
two-electron integrals in AO basis. Sξµν is the derivative of AO overlap
integral. ( )kPµνσ and ( )kWµνσ are
,
( ) ( ) ,C O
k kp pq q
pqP c P cµνα µ α α ν α≡ ∑ (132a)
,
( ) ( ) ,O V
k kp pq q
pqP c P cµνβ µ β β ν β≡ ∑ (132b)
( ) ( ) ,k kp pq q
p qW c W cµνα µ α α ν α
≤
≡ ∑ (133a)
( ) ( ) .k kp pq q
p qW c W cµνβ µ β β ν β
≤
≡ ∑ (133b)
６８
In addition, ( ),
kµνσ κλτΓ are given by
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ),
1= [2 ( ) ( )]2
k k k k k k k kH HP D c P D P D c X X X Xµνσ κλτ µνσ κλτ στ µκσ νλσ µλσ νκσ σα τβ µλ νκ µκ νλδ δ δΓ − + − +
i ( ) i ( ) i ( ) i ( )1 2 1 2s ( ) [( ) ( ) ( ) ( ) ntra k ntra k ntra k ntra k
H O V O V O V O Vgn k c X X X Xσα τβ µλ µλ κν κνδ δ+ × − −
i ( ) i ( ) i ( ) i ( )1 2 1 2( ) ( ) ( ) ( ) ntra k ntra k ntra k ntra k
CO CO CO COX X X Xµλ µλ κν κν+ − −
i ( ) i ( ) i ( ) i ( )1 2 2 1( ) ( ) ( ) ( )nter k nter k nter k nter k
CO O V CO O VX X X Xµν κλ µν κλ+ +
i ( ) i ( ) i ( ) i ( )2 1 1 2( ) ( ) ( ) ( )nter k nter k nter k nter k
O V CO O V COX X X Xµν κλ µν κλ+ +
i ( ) i ( ) i ( ) i ( )1 2 2 1( ) ( ) ( ) ( )nter k nter k nter k nter k
CO O V CO O VX X X Xµλ νκ µλ νκ− −
i ( ) i ( ) i ( ) i ( )2 1 1 2( ) ( ) ( ) ( ) ],nter k nter k nter k nter k
O V CO O V COX X X Xµλ νκ µλ νκ− − (134)
where
,
,C O
p pp
D c cµνα µ α ν α≡ ∑ (135a)
,C
p pp
D c cµνβ µ β ν β≡ ∑ (135b)
, ,
( ) ( ) ( ) ,C OO V
k k kp pq pq q
p qX c U X cµν µ α ν β≡ ∑∑ (136)
and
i ( ) ( )( ) ,O V
ntra k OmV kOmV p pq pq q
p qX c U X cµν µ α ν β≡ ∑∑ (137a)
i ( ) ( )( ) ,C O
ntra k COm kCOm p pq pq q
p qX c U X cµν µ α ν β≡ ∑∑ (137b)
i ( ) ( )( ) ,O V
nter k OmV kOmV p pq pq q
p qX c U X cµν µ α ν β≡ ∑∑ (137c)
６９
i ( ) ( )( ) .C O
nter k COm kCOm p pq pq q
p qX c U X cµν µ α ν β≡ ∑∑ (137d)
Finally, the gradient of the ground state energy is given by:
) ,,
= ( | ,E h D S W ξξ ξ ξµν µνσ µν µνσ µνσ κλτ
µνσ µνσ µνσ κλτ
µν κλ′ ′− + Γ∑ ∑ ∑ (138)
where
,1= ( ).2 HD D c D Dµνσ κλτ µνσ κλτ στ µλσ νκσδ′Γ − (139)
,
= p pq qp q
W c F cµνσ µ σ σ ν σ′ ∑ (140)
The gradient of excited response states can be obtained by adding the
gradient of excitation energy in Eq. (131) and that of ground energy in
Eq. (138). The second term on the right-hand side of Eq. (134) is
derived from the spin-pairing coupling in Eq. (107). Without this term,
all equations for the energy gradient are almost same as those of SF-
TDDFT.[46] Similarly, only the difference is using the expanded
excitation amplitudes, ( ) ( )k kpq pqU X , in MRSF-TDDFT. In the same manner,
one can simply utilize the same existing routines for SF-TDDFT.
７０
NUMERICAL RESULTS
In this section, it is investigated that how spin-contamination and spin-
pairing coupling affects various calculations containing 1. Vertical
excitation energy, 2. Singlet-triplet energy gap, 3. Geometry-
optimization structure, 4. Adiabatic excitation energy, 5. Minimum
energy conical intersection, 6. Non-adiabatic coupling matrix elements,
and 7. Non-adiabatic molecular dynamics.
MRSF-, MRSF(0)-, SF-TDDFT, and MRSF-CIS are utilized for these
calculations. The MRSF(0)-TDDFT is a MRSF-TDDFT without the
spin-pairing coupling and MRSF-CIS is a MRSF-TDDFT using 100%
HF exchange. All calculations are performed by a local development
version of the GAMESS-US program.[52]
The differences between MRSF(0)- and SF-TDDFT can specifically
represent the pure spin-contamination effect of SF-TDDFT. Also, the
effects of spin-pairing coupling to geometry can be understood from
the differences between MRSF- and MRSF(0)-TDDFT.
７１
1. Vertical excitation energy
1.1 P excited state of Be atom
Be atom has the 1S ground state arising in the configuration 1s22s2.
The low-lying excited states of Be are the 3Px,y,z and the 1Px,y,z states
arising in the configuration 1s22s12pk1, k=x,y,z. The SF-TDDFT and
the MR-SF-TDDFT calculations use the 3P z state (1s22s12pz
1) as the
reference state. With this choice of reference, the =0,M IS
LX and
=0,M ISRX amplitudes generate the 1Pz and 3Pz states, while the rest of
the P states are produced by the V OI
β αX and V O
I
α βX amplitudes.
The results of the calculations are collected in Tab. 4. SF-TDDFT,
MRSF-TDDFT(0) and MRSF-TDDFT, yield nearly identical energy for
the 1S ground electronic state since there is little spin-contamination
in the SF-TDDFT. The collinear SF-TDDFT method yields relatively
minor spin-contamination of the 1Pz and 3Pz components, for which the
excitation amplitudes arise in the O→O excitation (type I in Fig. 2).
However, the 3Px,y components, which are described by the amplitudes
arising in the OV excitation (type II in Fig. 2), are strongly spin-
contaminated; 2⟨ ⟩S =1.0, which is typical for mixtures of the true
７２
singlet and triplet states. As a result, there are absences of 1Px,y
components. Strictly, one cannot say the strongly spin-contaminated
states as neither singlet nor triplet states. Hence, an erroneous energy
splitting between the (x,y) components and z component of the 3P and
1P multiplets is predicted by SF-TDDFT; 0.811 eV and 1.247 eV,
respectively.
Although the removal of the spin-contamination by MRSF-TDDFT(0)
does not improve the energies of the multiplet components, the 1Px,y
and 3Px,y can now be distinguished. The use of the pairing strengths in
MRSF-TDDFT lifts the degeneracy of these components of the 1P and
3P multiplets, see the fourth column in Tab. 4. The magnitude of the
splitting between the components of the same multiplet with the non-
zero pairings is considerably reduced as compared to SF-TDDFT; the
3Px,y and 3Pz splitting is now 0.233 eV and the 1Px,y and 1Pz splitting is
reduced to 0.223 eV. It can be expected that the residual splitting
between the (x,y) and z components can be further reduced by
considering more accurate expressions for the pairing strengths than
simple Eq. (104) used here.
To understand the origin of the residual splitting, calculations with 100%
HF exchange instead of the XC functional (labeled MRSF-CIS in Tab.
７３
4) were performed. The use of the self-interaction free HF exchange
completely eliminates the erroneous splitting between the 3Px,y and 3Pz
components and reduces the 1Px,y–1Pz splitting to only 0.09 eV. This
indicates that the bulk of the splitting between the multiplet
components may be caused by the effect of the self-interaction error
of the density functional. The remaining tiny splitting between the
components of the 1P multiplet are probably caused by the
incompleteness of the mixed reference.
Not only for Be atom but also for other examples, MRSF-TDDFT
properly splits an unphysical state of SF-TDDFT (half-singlet and
half-triplet mixed state) into correct singlet and triplet states. A
posteriori spin-pairing coupling is expected to improve the split states.
７４
State SF-TDDFT MRSF-TDDFT(0) MRSF-TDDFT MRSF-CIS 1 S -14.651 (0.00) -14.651 (0.00) -14.651 (0.00) -14.584 (0.00) 3 Pz 2.877 (1.98) 2.899 (2.00) 2.900 (2.00) 2.107 (2.00)
3 Px,y 3.688 (1.00) 3.688 (2.00) 2.667 (2.00) 2.107 (2.00)
1 Pz 4.935 (0.02) 4.913 (0.00) 4.913 (0.00) 6.042 (0.00)
1 Px,y 3.688 (0.00) 4.690 (0.00) 5.952 (0.00)
Table 4. Ground state total energies (Hartree) and excitation energies (eV) for
Be atom. MRSF-TDDFT(0) denote MRSF-TDDFT without spin-pairing
coupling, and MRSF-CIS denote using 100% HF exchange within the MRSF-
TDDFT formalism. The BHHLYP functional and 6-31G basis set are utilized
for these calculations. The expectation value of S 2 is given in parentheses.
７５
1.2 Singlet valence excited state of molecule
Benchmark calculations for the singlet valence excited states with
organic molecules of Thiel set[53] were performed. Vertical
excitation energies to Franck-Condon excited states are calculated by
MRSF- and SF-TDDFT with cc-pVTZ basis set.[54] Mean absolute
errors (MAE) in unit of eV for the benchmark calculations by MRSF-,
SF-, and LR-TDDFT are tabulated in Tab. 5 compared against the
TBE-2 reference.[55] In MRSF-TDDFT, sets of MAE with different
the pairing-strength coefficient cSP are tabulated.
As the cSP coefficient increases from 0.3 to 0.9 the MAE values
somewhat rise. This is related to the fact that the vertical excitation
energies also rise, although weakly, as cSP changes. Note, the SF-
TDDFT MAE values can be considered as those obtained from MRSF-
TDDFT at cSP=0, due to the fact that MRSF-TDDFT is similar to SF-
TDDFT when the pairing-strength is neglected (cSP=0). Thus, one can
say that there is a minimum for the dependence of MAE on cSP. In other
word, there is a minimum value of MAE at a particular value(s) of the
cSP coefficient. For the B3LYP, PBE0, and BHHLYP this minimum is at
the following values of cSP: 0.4-0.5, 0.3-0.5, and 0.0-0.3,
correspondingly. This means that overall MRSF-TDDFT outperforms
７６
SF-TDDFT.
LR-TDDFT gives better MAE than MRSF- and SF-TDDFT for every
xc functional in this calculations. This can be understandable. Since
the vertical excitation energy is calculated at the optimized structure
in the S0 ground state, the reference state of LR-TDDFT (i.e., closed-
shell singlet state) describes the geometry better than that of MRSF-
and SF-TDDFT (i.e., open-shell triplet state). Better reference could
give better linear response of density, which lead to give better
vertical excitation energy.
Although the vertical excitation energy is important, the main focus of
SF- and MRSF-TDDFT is on describing conical intersections and
single-bond breaking/twisted systems. Therefore, we can conclude
that the slightly large 0.5 ~ 0.6 MAE at the Franck-Condon region can
be allowed for many other advantages around conical intersections and
avoided crossing points.
７７
Table 5. Mean absolute errors MAE (in eV) for the vertical excitation energies
computed at MRSF-, SF-, and LR-TDDFT level of theory compared against
the TBE-2 reference. Four different xc functionals and the cc-pVTZ basis
set are utilized for this benchmark calculations. In MRSF-TDDFT, sets of
MAE with different the pairing-strength coefficient cSP are given.
B3LYP PBE0 BHHLYP M08-SO
cSP MRSF-TDDFT
0.3 0.58 0.51 0.53 0.50
0.4 0.58 0.50 0.54 0.51
0.5 0.58 0.50 0.55 0.52
0.6 0.58 0.51 0.57 0.54
0.7 0.59 0.51 0.60 0.57
0.8 0.60 0.53 0.63 0.59
0.9 0.61 0.54 0.65 0.62
SF-TDDFT
0.65 0.57 0.53 0.51
LR-TDDFT
0.30 0.28 0.53 0.34
７８
2. Singlet-triplet energy gap
Singlet-triplet energy gap is an important chemical property to study
intersystem crossing or singlet fission. We have performed MRSF-
and LR-TDDFT calculations for the singlet-triplet (ST) gaps in a
series of molecules. For the majority of the molecules, adiabatic ST
gaps were calculated, except c-C4H4, C3H6, br-C7H14, ln-C7H14, C2H4,
butadiene, and hexatriene for which vertical ST gaps were calculated.
6-31G(d) basis set is utilized for all calculations with several other
functionals; D18X, B3LYP, PBE0, BHHLYP and M08-HX. Here, a
functional D18X functional stands for a new hybrid functional
consisting 0.15 Becke + 0.75 Hartree-Fock exchange and Lee-Yang-
Parr correlation, which yield the best performance for MRSF-TDDFT.
Figure 6a and 6b show absolute errors of the ST gaps with respect to
experimental results for MRSF- and LR-TDDFT, respectively. For
small molecules from NH to PH2+, the best agreement with the
experimental ST gaps is obtained with the use of the D18X functional.
The use of lower fraction of the HF exchange resulted in greater
errors. For molecules of medium size, it were the M08-HX and
BHHLYP functionals that showed the best agreement with the
experimental ST gaps.
７９
On the other hand, with LR-TDDFT, the best agreement with the
experiment is delivered by M08-HX. For the medium-sized molecules
all functionals give the deviations from the experiment strongly
exceeding 0.36 eV with large values for absolute errors.
８０
Figure 6. Comparison of the absolute errors (AE) of singlet-triplet energy gaps
for the organic molecules. 6-31G(d) basis set is utilized for all calculations.
８１
3. Geometry-optimization structure
In this subsection, it is investigated that how spin-
contamination and spin-pairing coupling affects the optimized
geometries of ground and excited states. All geometries were obtained
by using the analytic gradients of MRSF-, MRSF(0)-, and SF-TDDFT.
The BHHLYP[56-58] collinear XC kernel combined with the 6-31G (d)
basis set[59] were adopted for TDDFT calculations. In order to
measure accuracy of these geometries, geometry optimizations with
the equation-of-motion coupled-cluster singles and doubles (EOM-
CCSD)[60, 61] were also performed as a benchmarks level of theory
with the same basis set. Geometries of 8 organic molecules are
optimized for different states, which leads 20 different geometries (8
for S0 state, 8 for S1 state, and 4 for S2 state). All the structures were
optimized with the default threshold of 10-4 a.u./bohr and the default
integration grid for DFT (nrad = 96, nleb = 302) and TDDFT (nrad =
48, nleb = 110). All minimum points are confirmed by numerical
calculations of the Hessian.
After aligning all geometries obtained by different methods by
using vmd, geometric RMSDs of SF-, MRSF(0)- and MRSF-TDDFT
with respect to the reference geometry of EOM-CCSD are calculated.
８２
2
EOM-CCSD1
1RMSD .N
MmN =
= −∑ R R (141)
Here, N is the number of atoms for a molecule and MR is the
geometry vector of an optimized molecule by the M method (where M
=SF-, MRSF(0)-, or MRSF-TDDFT). The double vertical bar denotes
the Euclidean norm. Each RMSD is represented as a point in Fig. 7
along the 2⟨ ⟩S value of SF-TDDFT.
The spin contamination resulting from missing configurations
for the S0 states is rare. Thus, RMSDs of the S0 state are accumulated
around 0.0 ~ 0.1 of the 2⟨ ⟩S values. At the same time, their geometric
RMSDs against EOM-CCSD turned out to be small. The predicted
results with 2⟨ ⟩S values of 0.1 0.7 are coming from the type I
(O→O). Even for excited states composed of the spin-complete type I
(O→O) configurations of SF-TDDFT shown in Fig. 1, considerable spin
contamination appears due to the asymmetric nature of SF-TDDFT.
Both MRSF(0)- and MRSF-TDDFT well improve the geometric RMSDs
compared to those of SF-TDDFT. However, in the cases of
cyclopentadiene (S1), propanamide (S1), formamide (S1), and acetamide
(S1), MRSF-TDDFT is slightly inferior to MRSF(0)-TDDFT. While, the
MRSF-TDDFT in particular significantly improves RMSDs with
2 > 0.8⟨ ⟩S which are from the type II (C→O) and type III (O→V).
８３
Overall, the average RMSD of MRSF-TDDFT as represented
by horizontal black line is smaller than that of MRSF(0)-TDDFT in blue
line. The improved prediction accuracy of MRSF-TDDFT can justify
the introduction of a posteriori coupling.
８４
Figure 7. Geometric RMSDs of SF-, MRSF(0)- and MRSF-TDDFT with respect
to the reference geometry of EOM-CCSD. Each point denotes the RMSD for
an optimized structure for 8 organic molecules. There are 20 points (8 for S0,
8 for S1, and 4 for S2 optimized structures) The x-axis represents the
expectation value of S2 operator for SF-TDDFT. The red, blue and black
horizontal lines represent the average values of SF-, MRSF(0)- and MRSF-
TDDFT, respectively.
８５
4. Adiabatic excitation energy
In this subsection, adiabatic excitation energies (AEEs) were
calculated with the optimized structures obtained in previous
subsection. Then, the AEEs by SF-, MRSF(0)-, MRSF-TDDFT are
compared to those by EOM-CCSD and the magnitude of differences
are presented in Fig. 8 in unit of eV with the same x-axis as Fig. 7.
In the case of AEEs, the MRSF-TDDFT performs extremely
well to the degree that many of predicted AEEs are nearly identical to
those of EOM-CCSD, which is also seen from the average AEE
differences as presented by the black horizontal line. The improved
prediction accuracy of MRSF-TDDFT can justify the introduction of a
posteriori coupling.
８６
Figure 8. Magnitudes of AEE differences of SF-, MRSF(0)- and MRSF-TDDFT
with respect to the reference AEE of EOM-CCSD in eV. Each point denotes the AEE difference for the same set of Fig. 7. Thus, there are 12 points (8 for AEE(S1-S0) and 4 for AEE(S2-S0)) The x-axis and horizontal lines represent same as Fig. 7.
８７
5. Conical intersection
5.1 Minimum energy conical intersection of PSB3
The penta-2,4-dieniminium cation (PSB3) is an important model
system for vision used to test the ability of methods to correctly
describe topology of the conical intersections between the S0 and S1
states.[28, 29, 62-64] While LR-TDDFT fails to produce the conical
intersections correctly,[27-29, 65] SF-TDDFT does this correctly.
[28, 46, 65] However, 2⟨ ⟩S of S0 and S1 at the MECI geometry of SF-
TDDFT are 0.31 and 0.36, respectively.[42]
The MECI geometries of MRSF- and MRSF(0)-TDDFT also can be
obtained and the spin contamination is eliminated. Geometric RMSDs
of MRSF- and MRSF(0)-, SF-TDDFT with respect to MRCISD are
0.077, 0.165 and 0.240 in unit of Å, respectively. The MECI geometries
are aligned and presented in Fig. 9.
As clearly seen in the RMSDs as well as in Fig. 9, the geometric
improvement between SF- and MRSF(0)-TDDFT is seen, which is
purely from spin contamination. The geometry is further improved in
MRSF-TDDFT with a posteriori coupling.
８８
Figure 9. Aligned MECI geometry for the PSB3 molecule optimized by MRSF-,
MRSF(0)-, SF-TDDFT, and MR-CISD. The geometric RMSDs of MRSF- and
MRSF(0)-, SF-TDDFT with respect to MRCISD are 0.077, 0.165 and 0.240 in
Å, respectively.
８９
5.2 Topology of conical intersection for PSB3
In describing non-adiabatic process, the importance of the topology of
the conical intersection has been emphasized from the early stage of
study.[66-68] The conical intersection topology of PSB3 molecule has
been studied by Gozem et al.[28] and Huix-Rotllant et al..[29] They
reported that multi-reference methods (CASSCF and MRCISD) yield
double cone topology (i.e., conical intersection), while a linear S1/S0
crossing is obtained by the LR-TDDFT (i.e., linear intersection). In
addition, SF-TDDFT provided a correct topology even though there is
a spin-contamination.
This topology was able to be measured by calculating difference of
energies between S1 and S0 states in a circular loop around the conical
intersection for each method. The double cone topology should
provide non-zero energy difference anywhere in the loop, while the
linear topology yield two zero points in the loop.
We performed the loop calculation with the MRSF-TDDFT shown in
Fig. 10. MRSF-TDDFT predict the non-zero energy difference round
the loop; hence, the correct double cone topology is reproduced by
the method.
９０
Figure 10. Energy difference between S1 and S0 states calculated a loop around
the conical intersection of PSB3 molecule with MRSF-TDDFT.
９１
6. Non-adiabatic coupling term
6.1 Numerical non-adiabatic coupling term
The Tully’s fewest-switches trajectory-surface-hopping (TSH)
method is one of the most popular semi-classical methods to study
excited state dynamics with radiationless vibronic transition. In TSH
method,[69] nuclear wave packets are described by ensembles of
independent classical trajectories. Each trajectory is propagated on a
single adiabatic PES, and it is allowed to hop to other adiabatic PESs
at every time step according to hopping probabilities. The hopping
probabilities between different electronic states depend on the
nonadiabatic coupling term (NACT). The NACT can be obtained
numerically by the finite difference approximation in terms of overlap
integrals (OIs) between wave functions at different time steps as
( 2) ( 2)I Jt tt∂
Ψ − ∆ Ψ − ∆∂
( )1 ( ) ( ) ( ) ( ) .2 I J I Jt t t t′ ′≅ Ψ − ∆ Ψ − Ψ Ψ − ∆∆
(142)
Here, the notation prime is introduced to emphasize that the state is
in a different time-step t as opposed to t -∆. In this subsection, the
time variable will be omitted with the notation prime.
９２
The OIs between wave functions at different time steps are
represented as a function of products OIs between MOs at different
time steps.
( )1 2 1 21 2M MI J i i i i i M if ε φ φ φ φ φ φ′ ′ ′ ′Ψ Ψ = (143)
where 1 2 Mi i iε is the Levi-Civita symbol which is the sign of a
permutation of the natural number.
6.2 Fast overlap evaluations with truncation
If one multiplies an unit parameter λ, i.e., λ=1, to each OI between MOs,
mm iφ φ ′ , only when the index is different mm i≠ , the OIs between the
wave function can be represented as ascending order of the parameter
λ as
0
.M
mI J m
mA λ
=
′Ψ Ψ = ∑ (144)
Although the MOs at consecutive time steps are formally
nonorthogonal, they become nearly orthogonal as time-step size ∆
becomes zero. Thus, OIs between MOs, i.e., mm iφ φ ′ become nearly
the Kronecker delta function , mm iδ . When using a time step smaller
９３
than change of the wave function, especially in the non-adiabatic
molecular dynamics simulation, one can truncate the series depending
on an order of the parameter λ.[70]
The computing times for calculating an OI between wave functions are
obtained with different the truncation order from 0 to 2 and without
the truncation. Here, wave functions of SF-TDDFT are utilized and
size of basis set M is varying. The ratio of the computing times with
respect to the time with the zeroth order is shown in Fig. 11.
The truncation up to the second and the zeroth order gives four and
five orders improvements of system size compared to the calculation
without truncation, which is enormous time reduction for the OI
calculation.
９４
Figure 11. Ratios of computation times for evaluating OIs varying the truncation
order with respect to the time with the lowest truncation order. Both x- and y-axis are represented in log scale. From the slopes one can obtain the computational order of system size.
９５
6.3 Accuracy of truncation
In order to investigate accuracies of the truncated OI in actual
calculations, the NACTs near three different conical interactions were
studied. They are twisted-pyramidalized, H-migrated ethylene, and
twisted-pyramidalized stilbene. The BHHLYP[56-58] functional in
combination with 6-31G(d) basis set[59] was utilized. NACTs between
S1 and S0 states for the three different conical intersections and for
three different time-step sizes are tabulated in the third column of Tab.
6. NACTs errors with the truncation of OIs are tabulated in the fourth,
the fifth, the sixth columns. Those are calculated with truncated OIs
up to the 0th, 1st, 2nd order of the λ. All NACT values even with non-
truncated OIs are subjected to the finite difference approximation,
therefore any differences below 10-7 are meaningless. In the table,
when the error magnitudes become smaller than 10-7, we simply
specify them as < 10-7.
It can be expected that the NACT errors with truncated OIs up to the
0th, 1st, 2nd order of λ are nearly on the zeroth, the first, and the
second order of the time-step size, respectively. This expectation is
supported by results for which the error magnitudes with OIs up to the
2nd order is always under 10-7, and those up to the 1st order yield
９６
similar dependencies in the finite difference approximation. On the
other hand, those up to the 0th order are one or two orders larger than
others, and slightly larger than those of the finite difference
approximation. Nevertheless, these are still highly accurate compared
to the absolute NACT values. In general, it can be concluded that the
truncated OIs up to the 1st and 2nd order do not introduce additional
errors to the NACT. On the other hand, although the truncated OIs with
only 0th order adds error higher by one order of magnitude, it is still
capable of producing accurate values compared to the absolute NACT.
９７
Molecule Δ (fs) NACT (a.u.) NACT with truncation - NACT (a.u.)
w/o truncation 0th 1st 2nd
Twisted
ethylene
0.1 0.2361078 2.8 × 10-6 < 10-7 < 10-7
0.2 0.1199419 2.5 × 10-6 < 10-7 < 10-7
0.5 0.0480514 4.5 × 10-6 -6.0 × 10-7 < 10-7
H-
migrated
Ethylene
0.1 0.0885232 4.8 × 10-6 < 10-7 < 10-7
0.2 0.1156795 3.0 × 10-6 < 10-7 < 10-7
0.5 0.0472909 5.4 × 10-6 < 10-7 < 10-7
Twisted
stilbene
0.1 0.0105009 4.7 × 10-6 -1.2 × 10-7 < 10-7
0.2 0.0092561 5.2 × 10-6 -1.1 × 10-7 < 10-7
0.5 0.0085117 9.9 × 10-6 -2.1 × 10-7 < 10-7
Table 6. Errors of nonadiabatic coupling term (NACT) with the truncated
overlap integrals. NACTs (a.u.) between S1 and S0 states for three different conical intersections and for three different time-step sizes (Δ) are in the third column. NACTs error (a.u.) with the truncation of OIs are in the fourth, the fifth, the sixth columns. Those are calculated with truncated OIs up to the 0th, 1st, 2nd order of the λ. In all calculations, the NACTs were calculated between the given geometries and propagated geometries by velocity-Verlet algorithm with its gradient of the first excited state. “< 10-7” denotes that the magnitude of difference is lower than 1.0 × 10-7.
９８
7. Non-adiabatic molecular dynamic
7.1 Photoisomerization and photocyclization of cis-stilbene
The photoisomerization mechanism has been examined considerable
experimental and theoretical studies. As a prototypical model system,
1,2-diphenylethylene (stilbene) has been intensively studied. From
early experimental research, the quantum yield for the photoreaction
of the excitation on the S1 state of cis-stilbene have been reported to
be 10% for DHP, 35% for trans-stilbene and 55 % for cis-stilbene.
Figure 12[71] illustrate a schematic diagram for the excited (S1) cis-
stilbene dynamics based on previous mechanism studies. The three
branching points are well known represented by A, B and C,
respectively in Fig. 12. The S1 cis-stilbene branch into two channels
at the A point, which are referred as cis-trans (photoisomerization)
and cis-DHP (photocyclization) channels. As much as 70% of S1 cis-
stilbene head to cis-trans channel.[72, 73] At the B branching point,
half of them yield S0 trans-stilbene, and the other half convert to S0
cis-stilbene.[74-77] Meanwhile, the remaining 30% of S1 cis-stilbene
head to the cis-DHP channel, and one-third yield S0 DHP while the
other two-thirds become S0 cis-stilbene at the C point.[78]
９９
Figure 12. Schematic diagram of cis-stilbene photodynamics. Three branching
points A, B, and C are presented on the schematic potential energy diagram.
１００
7.2 Branching ratio and Quantum yield
Nonadiabatic molecular dynamics simulations for the ππ* excited cis-
stilbene have been performed for 2 ps using SF- or MRSF(0)-TDDFT
[50] with the TSH method.[69] Two sets of 50 trajectories are
simulated for the methods with initial geometries randomly chosen
from a 30 ps ground-state simulation of cis-stilbene. Full vibrational
degrees of freedom have been explored with 0.5 fs time step divided
into sub-time steps of 5.0 × 10-5 fs. The BHHLYP hybrid
functional[56-58] with 6-31G(d) basis set[59] has been utilized for
all calculations. The nonadiabatic coupling terms (NACT) are
numerically calculated with finite difference method and with truncated
overlap integrals[70] as described in previous subsection 5 of Non-
adiabatic coupling term.
First, we could obtain branching ratios at three branching points A, B
and C and quantum yield for both sets of calculations, and these are
tabulated in Tabs. 7 and 8. Error bars for all values denote the 90%
confidence interval obtained by the bootstrapping.
Branching ratios of the MRSF-TDDFT at branching points B and C are
well agree with those of experimental results except for slight
１０１
discrepancy at a branching point A as 70/30 for experiment and
60(±9)/40(±9) for MRSF-TDDFT. On the other hand, the branching ratios at
A and C for SF-TDDFT are well agree, while the branching ratios at B have
quite large discrepancy as 50/50 for experiment and 74(±10)/26(±10) for SF-
TDDFT.
For the final quantum yields of cis-, trans-stilbene and DHP in Tab. 8,
all values with MRSF-TDDFT are well agree with those of
experiments as 56(±9)/32(±8)/12(±6) for MRSF-TDDFT and 55/35/10
for experiments. On the other hand, SF-TDDFT underestimates
quantum yield of cis-stilbene and overestimates that of trans-stilbene
as 36(±9)/50(±9) for SF-TDDFT and 55/35 for experiments. In NAMD
with LR-TDDFT,[79] it cannot yield DHP product at all, and that of
cis-stilbene is overestimated as 66(±3) for LR-TDDFT and 55 for an
experiment.
１０２
A B C
cis-trans cis-DHP trans cis DHP Cis
MRSF-TDDFT 60(±9) 40(±9) 53(±12) 47(±12) 25(±12) 75(±12)
SF-TDDFT 68(±9) 32(±9) 74(±10) 26(±10) 44(±16) 56(±16)
Experiment 70 30 50 50 33 67
Table 7. Branching ratio (%) at three branching points during non-adiabatic
molecular dynamics of cis-stilbene. Values in parentheses denote the 90%
confidence interval obtained by the bootstrapping
１０３
cis trans DHP
MRSF-TDDFT 56 (± 9) 32 (± 8) 12 (± 6)
SF-TDDFT 36 (± 9) 50 (± 9) 14 (± 6)
LR-TDDFT 66 (± 3) 34 (± 3) 0
Experiment 55 35 10
Table 8. Quantum yields (%) of ππ* excited cis-stilbene. Values in parentheses
denote the 90% confidence interval obtained by the bootstrapping
１０４
7.3 Dynamics in the branching point
Further understanding about the branching ratio is investigated with
two dimensional histogram shown in Fig. 13. The x-axis is a bond
length (R) between two carbons in which a new single bond is formed
in DHP product, and the y-axis is a dihedral angle (DC=C) between two
bonds which connect the centered double bond and two phenyl rings.
The histograms for all 50 trajectories in the excited states are
presented in a and b for MRSF- and SF-TDDFT, respectively. The
histograms for 30 and 34 trajectories headed to the cis-trans channel
in the ground state is presented in c and d for MRSF- and SF-TDDFT
with points where hopping from S1 to S0 occur. Likewise, the ground
state histograms for 20 and 16 trajectories headed to cis-DHP channel
are shown in e and f for MRSF- and SF-TDDFT, respectively with the
hopping points.
A clear discrepancy between S1 state histograms for MRSF- and SF-
TDDFT is seen. The A branching point is highly populated, while both
A and B points are highly populated in SF-TDDFT. It represent
trajectories hopping from S1 to S0 quite fast not to reside around B
points for MRSF-TDDFT. Since a difference between MRSF(0)- and
SF-TDDFT is the absence or presence of spin contamination, it is
１０５
expected that the difference come from spin contamination problem of
SF-TDDFT. Figure 14 shows averaged <S2> value of SF-TDDFT in
the excited state. Overall region for A and B points has 0.2 ~ 0.4 of
<S2> value, which is not negligible spin contamination.
Around branching point B (cis-trans), surface hoppings appear in
broad region of the histogram. It is seen that trajectories hopping with
relatively small R and DC=C become cis-stilbene, while these become
trans-stilbene in the other case. In MRSF-TDDFT, more trajectories
are hopping with relatively small R and DC=C compared to those of SF-
TDDFT. It lead to good agreement of branching ratio of MRSF-TDDFT,
while overestimate trans-stilbene branching ratio of SF-TDDFT as
74(±10) for SF-TDDFT and 50 for experiment.
Branching ratios at branching point C (cis-DHP) for both MRSF- and
SF-TDDFT are agree with experiment results shown in Tab. 7.
However, a difference between distributions of hopping points for two
methods are seen. Surface hoppings appear in narrow region of the
histogram for MRSF-TDDFT, while these are relatively scattered for
SF-TDDFT. In SF-TDDFT, there are a few hoppings with R less than
1.7 angstrom, which lead to DHP product. In Fig. 14, there is severe
spin contamination where <S2> > 0.4. Although the branching ratio of
１０６
SF-TDDFT is agree with experiment result, these are seemed to be
problematic hoppings.
１０７
Figure 13. Two-dimensional histogram of cis-stilbene NAMD dynamics
simulation. The histograms for 50 trajectories in excited states are presented
in a and b, The geometric RMSDs of MRSF- and MRSF(0)-, SF-TDDFT with
respect to MRCISD are 0.077, 0.165 and 0.240 in Å, respectively.
１０８
Figure 14. Averaged <S2> value of SF-TDDFT for 50 trajectories in excited
states.
１０９
CONCLUSION
A new method in the context of collinear spin-flip linear response
TDDFT is proposed, which employs an equiensemble of the = 1SM +
and = 1SM − components of the triplet state as a (mixed) reference
state. The TD-KS equation with the mixed state can be solved within
linear response formalism by the use of spinor-like open-shell orbitals.
It is a novel attempt to add configurations within the realm of TDDFT.
The resulting MRSF-TDDFT has several advantages over the
conventional collinear SF-TDDFT. MRSF-TDDFT gives more
accurate results than SF-TDDFT by eliminating the spin contamination
of the response states of SF-TDDFT. The accuracy of MRSF-TDDFT
has been tested and verified in various calculations. The spin
contaminated P state of Be atom for SF-TDDFT properly splits into
singlet (1P) and triplet states (3P). A posteriori coupling is shown to be
improve degeneracy of the P states. In the vertical excitation energy
calculations to Franck-Condon states for organic molecules, results of
MRSF-TDDFT give slightly better than those of SF-TDDFT compared
to reference results of the theoretical best estimate (TBE-2). For the
structures and adiabatic excitation energies obtained from geometry
optimization of organic molecules, MRSF(0)-TDDFT results are closer
１１０
than SF-TDDFT results compared to the EOM-CCSD results. It is
shown that MRSF-TDDFT results (with a posteriori coupling) further
improve the results. Comparing MRCISD calculation, MRSF(0)-TDDFT
is also superior than SF-TDDFT in calculations for the minimum
energy conical intersection of PSB3 molecule, which is the important
model system in vision. Likewise, a posteriori coupling improves the
result. Apart from the accuracy of MRSF-TDDFT, we propose a
truncation method to calculate non-adiabatic coupling which
drastically reduce computation time while maintaining greatly high
accuracy. Using this non-adiabatic coupling term, non-adiabatic
molecular dynamics of cis-stilbene molecules were able to be
performed by SF- and MRSF(0)-TDDFT. Both methods are superior
than LR-TDDFT since the methods produced a DHP product which
LR-TDDFT could not produce. The SF-TDDFT yield a non-negligible
amount of spin-contamination throughout the dynamics simulations
and the MRSF(0)-TDDFT could completely eliminate the spin
contamination. Branching ratio and quantum yield of MRSF(0)-TDDFT
is more closer to the experimental value than those of SF-TDDFT.
Not only for the accuracy but clear separation of singlet and triplet
response states considerably simplify the means by which excited
states are identified, especially in ``black-box''-type applications,
such as automatic geometry optimization, reaction path following, and
１１１
molecular dynamics simulations.
From these reasons it is highly expected that MRSF-TDDFT yield
improved results than SF-TDDFT by eliminating spin contamination
problem in general cases. Therefore, we can conclude that MRSF-
TDDFT has advantages over SF-TDDFT in terms of both practicality
and accuracy.
１１２
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１２４
ABSTRACT IN KOREAN (국문초록)
혼합참조 스핀젖힘 시간의존 밀도범함수 이론(MRSF-TDDFT)이 제안
됐다. 이 새로운 양자화학 계산 방법은 잘 알려진 시간의존 Kohn-Sham
방정식의 선형응답 이론에 혼합참조라는 새로운 개념을 도입해 유도된다.
혼합참조는 삼중항의 두 구성요소(MS=+1, -1)를 결합한 참조상태로, 혼
합참조로부터 나오는 선형응답은 기존 SF-TDDFT 방법이 만들지 못해
문제가 됐던 전자구성을 TDDFT 이론 내에서 생성한다. MRSF-TDDFT
의 핵 좌표에 대한 응답상태의 에너지 기울기 수식이 유도됐고, 프로그
램화 됐다. 새로운 방법의 에너지 그리고 에너지 기울기에 대한 계산 요
구량은 기존 SF-TDDFT 의 요구량과 거의 동일하다. 새롭게 제안 된
MRSF-TDDFT 방법은 기존 SF-TDDFT에 비해 실용성과 정확성 측면
에서 장점이 있다. 응답상태들을 단일항 또는 삼중항 상태로 완전히 분
리해냄으로써, 기존 SF-TDDFT의 응답상태들의 스핀오염문제를 제거한
다. 따라서, 특정 응답상태에 대한 자동 구조 최적화, 반응 경로 추적 또
는 분자 동력학 시뮬레이션과 같은 "블랙 박스(black-box)" 유형의 응용
에서 필수적인 응답 상태의 식별을 상당히 단순화한다. 또한, MRSF-
TDDFT의 정확성은 수직 여기 에너지, 단열 여기 에너지, 구조최적화
된 구조, 최소 에너지 원뿔 교차점, 비단열 상호작용 항 및 비단열 분자
동력학 시뮬레이션과 같은 다양한 방법으로 시험되고 검증되었다. 따라
서, MRSF-TDDFT 방법은 광화학 반응 연구에 유망한 양자화학 계산방
법으로 기대된다.
